Condensed Matter Physics

1 The nature of condensed matter
Condensed matter physics is the study of large numbers of atoms and molecules that are
“stuck together.” Solids and liquids are examples. In the condensed state many molecules
interact with each other. The physics of such a system is quite different from that of the
individual molecules because of collective effects: qualitatively new things happen because
there are many interacting particles. The behavior of most of the objects in our everyday
experience is dominated by collective effects. Examples of materials where such effects are
important are crystals and magnets.
This is a vast field: the subject matter could be taken to include traditional solid state
physics (basically the study of the quantum mechanics of crystalline matter), magnetism,
fluid dynamics, elasticity theory, the physics of materials, aspects of polymer science, and
some biophysics. In fact, condensed matter is less a field than a collection of fields with
some overlapping tools and techniques. Any course in this area must make choices. This is
my personal choice.
In this chapter I will discuss orders of magnitude that are important, review ideas from
quantum mechanics and chemistry that we will need, outline what holds condensed matter
together, and discuss how order arises in condensed systems. The discussion here will be
qualitative. Later chapters will fill in the details.
1.1 Some basic orders of magnitude
To fix our ideas, consider a typical bit of condensed matter, a macroscopic piece of solid
copper metal. As we will see later it is best to view the system as a collection of cuprous
(Cu+) ions and conduction electrons, one per atom, that are free to move within the metal.
We discuss some basic scales that will be important for understanding the physics of this
piece of matter.
LengthsAcharacteristic length that will be important is the distance between the Cu atoms.
In a solid this distance will be of order of a chemical bond length:
L ≈ 3 Å ≈ 3 × 10−8cm. (1.1)
Note that this is very tiny on the macroscopic scale. The whole art of condensed matter
physics consists in bridging the gap between the atomic scale and the macroscopic properties
of condensed matter.
Energies We can ask about the characteristic energy scales for the sample. One important
energy scale is the binding energy of the material per atom.Aclosely related quantity is the
melting temperature in energy units:
1357 K = 0.11 eV. (1.2)
This is a typical scale to break up the material. If we probe at much larger energies (KeV, for
example) we will be probing the inner shells of Cu, namely the domain of atomic physics,
or at MeV, the Cu nucleus, i.e. nuclear physics.
Cu has an interesting color (it is copper colored, in fact), so we might expect something
interesting at the scale of the energy of ordinary light, namely,
E ≈
ωopt ∼ 3 eV (1.3)
which is also the strength of a typical chemical bond. A somewhat larger, but comparable
scale is that of the Coulomb interaction of two electrons a distance L apart:
E ≈ e2/L ≈ 5 eV. (1.4)
These energies are low even for atomic physics. This means that in our study of condensed
matter we will always be interested only in the outer (valence electrons) which are least
bound.
SpeedsWhen a piece of Cu carries an electrical current of density, j, the conduction electrons
move at a drift velocity vd:
j = ne vd (1.5)
where n is the number density of conduction electrons and e is the charge on the electron.
For ordinary sized currents we find a very small speed, vd ≈ 0.01 cm/sec.
There is another characteristic speed, the mean thermal speed, vT of the Cu ions when they
vibrate at finite temperature.We estimate vT as follows. From the Boltzmann equipartition
theorem the mean kinetic energy of an ion is:
Mv2T/2 ∼ kBT. (1.6)
Here T is the absolute temperature, kB is Boltzmann’s constant, M is the mass of a Cu ion,
and vT is the mean thermal velocity. At room temperature we get vT ∼ 3 × 104 cm/sec.
There is a larger speed associated with the electrons, namely the quantum mechanical
speed of the valence electrons.We estimate this speed as [frequency of an optical transition]
x length:
v ∼ (E/
)(L) ≈ 107 cm/sec. (1.7)
As we will see below, there is another relevant speed, the magnitude of the Fermi velocity,
which is of the same order.
In any case, all of these speeds are small compared to the speed of light. Thus, we seldom
need the theory of relativity in condensed matter physics. (An exception is the spin-orbit
interaction of heavy elements.)
Large numbers and collective effects The essential point of the subject is that we deal
with very large numbers of ions and electrons,≈ 1027 in a macroscopic sample. In a famous
essay P. W. Anderson (1972) pointed out the significance of this fact. When many things
interact we often generate new phenomena, sometimes called emergent phenomena. Or,
as Anderson put it, “more is different.” Some examples of collective effects that we will
emphasize in this book are the existence of order of various types, e.g. crystalline order,
magnetic order, and superconducting order.
1.2 Quantum or classical
We have seen that we are interested in non-relativistic physics. We can go further: for the
case of Cu there are conduction electrons and Cu+ ions. What type of physics is applicable
to each? In particular, do we need quantum mechanics?Auseful criterion is to compare the
de Broglie wavelength of the relevant particle, λ = h/mv, to the interparticle spacing.
For the ions, the relevant speed is vT which we estimated above. Thus:
λ = h/(2MkBT)1/2 ≈ 10−9 cm << L. (1.
This is smaller than the spacing by an order of magnitude. For all ions in solids (except for
He and H at very low temperatures) we can use classical mechanics. (As we will see, for
vibrations of ions at low T, we need quantum mechanics too.)
For the electrons the situation is different because the electron mass, m, is is 63×1800
times smaller than the mass of a Cu ion, so we get
λ = h/(2mkBT)1/2 ≈ 3 × 10−7 cm >> L. (1.9)
Electrons are quantum mechanical for all temperatures.
1.3 Chemical bonds
Matter condenses because atoms and molecules attract one another. In the condensed state
they are connected by chemical bonds. This is the “glue” that holds condensed matter
together. We will summarize here some notions from chemistry which we will need in the
sequel.
van derWaals’ bonds At long ranges the dominant interaction between neutral atoms or
molecules is the van der Waals interaction which arises from the interaction of fluctuating
induced dipoles. For two neutral molecules (or atoms) a distance d apart this effect gives
rise to a potential energy of interaction given by:
V(r) ∼ −1/r6. (1.10)
This equation is universally true if the molecules are far apart compared to the size of
of their electronic clouds. For closed shell atoms and molecules such as Ar and H2 that
do not chemically react, the van der Waals’ interaction is the attractive force that causes
condensation. Since this is a weak, short-range force, materials bound this way usually have
low melting points.
A rough argument for the r−6 dependence is as follows: suppose there is a fluctuation
(a quantum fluctuation, in fact) on one of two molecules so that an instantaneous dipole
moment, p1, arises. This gives rise to an electric field of order E ∼ p1/d3 at the other
molecule. This electric field polarizes the other atom. To understand this, we introduce a
concept that we will use later, the polarizability, α, of the molecule. It is defined by:
pind = αE, (1.11)
where pind is the induced dipole moment. Note that in our system of units the polarizability,
α, has units of volume. It is roughly the molecular volume. Thus p2 ∼ αp1/d3. This finally
gives for identical molecules the fluctuating dipole-dipole interaction:
V ∼ p1p2/d3 ∼ αp2
1/d6. (1.12)
Since this expression depends on p2
1 there is a time-averaged value for the potential. It is
easy to show that the dipoles will be antiparallel so that the interaction is attractive. An
actual calculation of the coefficient of r−6, that is, of the average of p2
1, can be done (in
simple cases) using quantum mechanical perturbation theory.
Ionic bonds The chemistry of the valence electrons in a compound can lead to charge
transfer, e.g.:
Na + Cl → Na+Cl−. (1.13)
In this case there will be strong forces due to the charges, and the ions will be bound by the
Coulomb interaction:
V(r) = Zq1q2/r.
This is called ionic binding. Solid NaCl, table salt, is bound in this way. Ionic solids often
have very large binding, and very large melting points.
Covalent bonds In elements with s and p electrons in the outer shell, covalent sp3 orbitals
give rise to directed bonds where electrons between ions glue together the material. Semiconductors
such as Si, Ge, are bonded this way, as well as polymers and many biological
materials. There are intermediate cases between the covalent and ionic materials, such as
III-V semiconductors like GaAs.
Hydrogen bonds These arise in materials that containHsuch as ice. The proton participates
in the bonding. This is very important in biological materials.
Metallic bonding For most light metals like Cu or Na, the outer valence electrons are
delocalized for quantum mechanical reasons which we will discuss in great detail, later.
The electrons act as glue by sitting between the positively charged ions. These essentially
free electrons give rise to the electrical conduction of metals such as Cu.
1.4 The exchange interaction
We have talked about bonds between atoms in terms of spatial degrees of freedom of the
electrons, but we have not mentioned electron spin. There is another effect, very important
for magnetism, which arises from the interplay between the Pauli exclusion principle, the
spin degrees of freedom, and the electrostatic repulsion of electrons. It occurs, for example,
for atoms which have unpaired spins.
We recall from quantum mechanics that the Pauli principle says that electron wavefunctions
must be antisymmetric in the exchange of any two electrons. This implies that when
we bring two atoms together the many-electron wavefunction must vanish when two electrons
with parallel spins are at the same position. Therefore electrons with parallel spins
are likely to be farther apart in space than antiparallel ones, and therefore have a smaller
electrostatic repulsion. As a result, if the two atoms have parallel spins the energy is lower.
Thus spins and therefore magnetic moments tend to line up when electrons from adjacent
atoms overlap. This is called the exchange interaction. This is discussed in considerable
detail below, Section 9.2.1, or in standard texts on quantum mechanics, e.g. (Landau &
Lif***z 1977, Schiff 1968, Baym 1990).
There are a few comments we should make about this. One is that there needs to be
overlap of wavefunctions to have the effect work. The difference in energy between states
with parallel and antiparallel spins on adjacent atoms (the strength of the interaction) is
dependent on the overlap; the exchange interaction is very short range. Also, the size of the
energy difference is basically the electrostatic energy of two electrons an atomic distance
apart, a few electron volts.
Spin and symmetry effects need not favor parallel spins; it depends on the nature of
the wavefunctions and what energies are most important. A simple example of favoring
antiparallel spins is the hydrogen molecule, two electrons and two protons. In one
approach to the problem (the Heitler–London approximation) we build up the wavefunction
for the molecule from atomic wavefunctions centered on each proton. We can
then form symmetric and antisymmetric combinations of these functions, as above. However,
since the total wavefunction must be antisymmetric, parallel electron spins (total
spin 1) go with the antisymmetric spatial function, and antiparallel spins (total spin 0)
go with the symmetric spatial function; for more details see (Baym 1990). The electrostatic
interaction with the hydrogen nuclei favors the symmetric state since the electrons
spend more time between the nuclei, and the kinetic energy of the symmetric state is
lower. As a result the ground (bonding) state of H2 has total spin 0, and is symmetric in
space.

Quantum Field Theoryof Many-body Systems
From the Origin of Soundto an Origin of Light and Electrons
Xiao-Gang Wen
Department of Physics, MIT

PREFACE
The quantum theory of condensed matter (i.e. solids and liquids) has been dominated
by two main themes. The first one is band theory and perturbation theory. It
is loosely based on Landau's Fermi liquid theory. The second theme is Landau's
symmetry-breaking theory and renormalization group theory. Condensed matter
theory is a very successful theory. It allows us to understand the properties of
almost all forms of matter. One triumph of the first theme is the theory of semiconductors,
which lays the theoretical foundation for electronic devices that make
recent technological advances possible. The second theme is just as important. It
allows us to understand states of matter and phase transitions between them. It is
the theoretical foundation behind liquid crystal displays, magnetic recording, etc.
As condensed matter theory has been so successful, one starts to get a feeling
of completeness and a feeling of seeing the beginning of the end of condensed
matter theory. However, this book tries to present a different picture. It advocates
that what we have seen is just the end of the beginning. There is a whole new world
ahead of us waiting to be explored.
A peek into the new world is offered by the discovery of the fraction quantum
Hall effect (Tsui et al, 1982). Another peek is offered by the discovery of high-Tc
superconductors (Bednorz and Mueller, 1986). Both phenomena are completely
beyond the two themes outlined above. In last twenty years, rapid and exciting
developments in the fraction quantum Hall effect and in high-Tc superconductivity
have resulted in many new ideas and new concepts. We are witnessing an emergence
of a new theme in the many-body theory of condensed matter systems. This
is an exciting time for condensed matter physics. The new paradigm may even
have an impact on our understanding of fundamental questions of nature.
It is with this background that I have written this book.1 The first half of this
book covers the two old themes, which will be called traditional condensed matter
theory.2 The second part of this book offers a peek into the emerging new theme,
which will be called modern condensed matter theory. The materials covered in
the second part are very new. Some of them are new results that appeared only a
few months ago. The theory is still developing rapidly.
After reading this book, I hope, instead of a feeling of completeness, readers
will have a feeling of emptiness. After one-hundred years of condensed matter theory,
which offers us so much, we still know so little about the richness of nature.
However, instead of being disappointed, I hope that readers are excited by our
incomplete understanding. It means that the interesting and exciting time of condensed
matter theory is still ahead of us, rather than behind us. I also hope that
readers will gain a feeling of confidence that there is no question that cannot be
answered and no mystery that cannot be understood. Despite there being many
mysteries which remain to be understood, we have understood many mysteries
which initially seemed impossible to understand. We have understood some fundamental
questions that, at the beginning, appeared to be too fundamental to even
have an answer. The imagination of the human brain is also boundless.3
This book was developed when I taught the quantum many-body physics course
between 1996 and 2002 at MIT. The book is intended for graduate students who
are interested in modern theoretical physics. The first part (Chapters 2-5) covers
traditional many-body physics, which includes path integrals, linear responses,
the quantum theory of friction, mean-field theory for interacting bosons/fermions,
symmetry breaking and long-range order, renormalization groups, orthogonality
catastrophe, Fermi liquid theory, and nonlinear -models. The second part (Chapters
6-10) covers topics in modern many-body physics, which includes fractional
quantum Hall theory, fractional statistics, current algebra and bosonization, quantum
gauge theory, topological/quantum order, string-net condensation, emergent
gauge-bosons/fermions, the mean-field theory of quantum spin liquids, and twoor
three-dimensional exactly soluble models.
Most of the approaches used in this book are based on quantum field theory
and path integrals. Low-energy effective theory plays a central role in many of our
discussions. Even in the first part, I try to use more modern approaches to address
some old problems. I also try to emphasize some more modern topics in traditional
condensed matter physics. The second part covers very recent work. About half of
it comes from research work performed in the last few years. Some of the second
part is adapted from my research/review papers (while some research papers were
adapted from parts of this book).
The book is written in a way so as to stress the physical pictures and to stress the
development of thoughts and ideas. I do not seek to present the material in a neat
and compact mathematical form. The calculations and the results are presented
in a way which aims to expose their physical pictures. Instead of sweeping ugly
assumptions under the rug, I try to expose them. I also stress the limitations of
some common approaches by exposing (instead of hiding) the incorrect results
obtained by those approaches.
Instead of covering many different systems and many different phenomena,
only a few simple systems are covered in this book. Through those simple systems,
we discuss a wide range of physical ideas, concepts, and methods in condensed
matter theory. The texts in smaller font are remarks or more advanced topics,
which can be omitted in the first reading.
Another feature of this book is that I tend to question and expose some
basic ideas and pictures in many-body physics and, more generally, in theoretical
physics, such as 'what are fermions?', 'what are gauge bosons?', the idea of
phase transition and symmetry breaking, 'is an order always described by an order
parameter?', etc. Here, we take nothing for granted. I hope that those discussions
will encourage readers to look beyond the nice mathematical formulations that
wrap many physical ideas, and to realize the ugliness and arbitrariness of some
physical concepts.
As mathematical formalisms become more and more beautiful, it is increasingly
easy to be trapped by the formalism and to become a 'slave' to the formalism.
We used to be 'slaves' to Newton's laws when we regarded everything as a collection
of particles. After the discovery of quantum theory,4 we become 'slaves'
to quantum field theory. At the moment, we want to use quantum field theory
to explain everything and our education does not encourage us to look beyond
quantum field theory.
However, to make revolutionary advances in physics, we cannot allow our
imagination to be trapped by the formalism. We cannot allow the formalism to
define the boundary of our imagination. The mathematical formalism is simply a
tool or a language that allows us to describe and communicate our imagination.
Sometimes, when you have a new idea or a new thought, you might find that you
cannot say anything. Whatever you say is wrong because the proper mathematics
or the proper language with which to describe the new idea or the new thought
have yet to be invented. Indeed, really new physical ideas usually require a new
mathematical formalism with which to describe them. This reminds me of a story
about a tribe. The tribe only has four words for counting: one, two, three, and
many-many. Imagine that a tribe member has an idea about two apples plus two
apples and three apples plus three apples. He will have a hard time explaining
his theory to other tribe members. This should be your feeling when you have a
truly new idea. Although this book is entitled Quantum field theory of many-body
systems, I hope that after reading the book the reader will see that quantum field
theory is not everything. Nature's richness is not bounded by quantum field theory.
I would like to thank Margaret O'Meara for her proof-reading of many chapters
of the book. I would also like to thank Anthony Zee, Michael Levin, Bas
Overbosch, Ying Ran, Tiago Ribeiro, and Fei-Lin Wang for their comments and
suggestions. Last, but not least, I would like to thank the copy-editor Dr. Julie
Harris for her efforts in editing and polishing this book.
Lexington, MA Xiao-Gang Wen
October, 2003

1
INTRODUCTION
1.1 More is different
• The collective excitations of a many-body system can be viewed as particles.
However, the properties of those particles can be very different from the
properties of the particles that form the many-body system.
• Guessing is better than deriving.
• Limits of classical computing.
• Our vacuum is just a special material.
A quantitative change can lead to a qualitative change. This philosophy is
demonstrated over and over again in systems that contain many particles (or many
degrees of freedom), such as solids and liquids. The physical principles that govern
a system of a few particles can be very different from the physical principles
that govern the collective motion of many-body systems. New physical concepts
(such as the concepts of fermions and gauge bosons) and new physical laws and
principles (such as the law of electromagnetisnl) can arise from the correlations of
many particles (see Chapter 10).
Condensed matter physics is a branch of physics which studies systems of many
particles in the 'condensed' (i.e. solid or liquid) states. The starting-point of current
condensed matter theory is the Schrodinger equation that governs the motion of a
number of particles (such as electrons and nuclei). The Schrodinger equation is
mathematically complete. In principle, we can obtain all of the properties of any
many-body system by solving the corresponding Schrodinger equation.
However, in practice, the required computing power is immense. In the 1980s,
a workstation with 32 Mbyte RAM could solve a system of eleven interacting electrons.
After twenty years the computing power has increased by 100-fold, which
allows us to solve a system with merely two more electrons. The computing power
required to solve a typical system of 1023 interacting electrons is beyond the imagination
of the human brain. A classical computer made by all of the atoms in our
universe would not be powerful enough to handle the problem/' Such an impossible
computer could only solve the Schrodinger equation for merely about 100
particles.6 We see that an generic interacting many-body system is an extremely
complex system. Practically, it is impossible to deduce all of its exact properties
from the Schrodinger equation. So, even if the Schrodinger equation is the correct
theory for condensed matter systems, it may not always be helpful for obtaining
physical properties of an interacting many-body system.
Even if we do get the exact solution of a generic interacting many-body system,
very often the result is so complicated that it is almost impossible to understand
it in full detail. To appreciate the complexity of the result, let us consider a tiny
interacting system of 200 electrons. The energy eigenvalues of the system are distributed
in a range of about 200 eV. The system has at least 2200 energy levels. The
level spacing is about 200 eV/2200 = 10-60 eV. Had we spent a time equal to the
age of the universe in measuring the energy, then, due to the energy-time uncertainty
relation, we could only achieve an energy resolution of order 10-33 eV. We
see that the exact result of the interacting many-body system can be so complicated
that it is impossible to check its validity experimentally in full detail.7 To really
understand a system, we need to understand the connection and the relationship
between different phenomena of a system. Very often, the Schrodinger equation
does not directly provide such an understanding.
As we cannot generally directly use the Schrodinger equation to understand an
interacting system, we have to start from the beginning when we are faced with a
many-body system. We have to treat the many-body system as a black box, just
as we treat our mysterious and unknown universe. We have to guess a low-energy
effective theory that directly connects different experimental observations, instead
of deducing it from the Schrodinger equation. We cannot assume that the theory
that describes the low-energy excitations bears any resemblance to the theory that
describes the underlying electrons and nuclei.
This line of thinking is very similar to that of high-energy physics. Indeed,
the study of strongly-correlated many-body systems and the study of high-energy
physics share deep-rooted similarities. In both cases, one tries to find theories
that connect one observed experimental fact to another. (Actually, connecting one
observed experimental fact to another is almost the definition of a physical theory.)
One major difference is that in high-energy physics we only have one 'material'
(our vacuum) to study, while in condensed matter physics there are many different
materials which may contain new phenomena not present in our vacuum (such
as fractional statistics, non-abelian statistics, and gauge theories with all kinds of
gauge groups).

1.2 'Elementary' particles and physics laws are emergent phenomena
• Emergence—the first principle of many-body systems.
• Origin of 'elementary' particles.
• Origin of the 'beauty' of physics laws. (Why nature behaves reasonably.)
Historically, in our quest to understand nature, we have been misled by a fundamental
(and incorrect) assumption that the vacuum is empty. We have (incorrectly)
assumed that matter placed in a vacuum can always be divided into smaller parts.
We have been dividing matter into smaller and smaller parts, trying to discover
the smallest 'elementary' particles—the fundamental building block of our universe.
We have been believing that the physics laws that govern the 'elementary'
particles must be simple. The rich phenomena in nature come from these simple
physics laws.
However, many-body systems present a very different picture. At high energies
(or high temperatures) and short distances, the properties of the many-body
system are controlled by the interaction between the atoms/molecules that form
the system. The interaction can be very complicated and specific. As we lower
the temperature, depending on the form of the interaction between atoms, a crystal
structure or a superfluid state is formed. In a crystal or a superfluid, the only
low-energy excitations are collective motions of the atoms. Those excitations are
the sound waves. In quantum theory, all of the waves correspond to particles, and
the particle that corresponds to a sound wave is called a phonon.8 Therefore, at
low temperatures, a new 'world' governed by a new kind of particle—phonons—
emerges. The world of phonons is a simple and 'beautiful' world, which is very
different from the original system of atoms/molecules.
Let us explain what we mean by 'the world of phonons is simple and beautiful'.
For simplicity, we will concentrate on a superfluid. Although the interaction
between atoms in a gas can be complicated and specific, the properties of emergent
phonons at low energies are simple and universal. For example, all of the phonons
have an energy-independent velocity, regardless of the form of the interactions
between the atoms. The phonons pass through each other with little interaction
despite the strong interactions between the atoms. In addition to the phonons, the
superfluid also has another excitation called rotons. The rotons can interact with
each other by exchanging phonons, which leads to a dipolar interaction with a force
proportional to 1/r4. We see that not only are the phonons emergent, but even the
physics laws which govern the low-energy world of the phonons and rotons are
emergent. The emergent physics laws (such as the law of the dipolar interaction
and the law of non-interacting phonons) are simple and beautiful.
8
I regard the law of 1/r4 dipolar interaction to be beautiful because it is not
1/r3, or 1/r4-13, or one of billions of other choices. It is precisely 1/r4, and so
it is fascinating to understand why it has to be 1/r4. Similarly, the 1/r2 Coulomb
law is also beautiful and fascinating. We will explain the emergence of the law of
dipolar interaction in superfluids in the first half of this book and the emergence of
Coulomb's law in the second half of this book.
If our universe itself was a superfluid and the particles that form the superfluid
were yet to be discovered, then we would only know about low-energy phonons.
It would be very tempting to regard the phonon as an elementary particle and the
1/r4 dipolar interaction between the rotons as a fundamental law of nature. It is
hard to imagine that those phonons and the law of the 1/r4 dipolar interaction
come from the particles that are governed by a very different set of laws.
We see that in many-body systems the laws that govern the emergent lowenergy
collective excitations are simple, and those collective excitations behave
like particles. If we want to draw a connection between a many-body system and
our vacuum, then we should connect the low-energy collective excitations in the
many-body system to the 'elementary' particles (such as the photon and the electron)
in the vacuum. But, in the many-body system, the collective excitations are
not elementary. When we examine them at short length scales, a complicated nonuniversal
atomic/molecular system is revealed. Thus, in many-body systems we
have collective excitations (also called quasiparticles) at low energies, and those
collective excitations very often do not become the building blocks of the model at
high energies and short distances. The theory at the atomic scale is usually complicated,
specific, and unreasonable. The simplicity and the beauty of the physics
laws that govern the collective excitations do not come from the simplicity of the
atomic/molecular model, but from the fact that those laws have to allow the collective
excitations to survive at low energies. A generic interaction between collective
excitations may give those excitations a large energy gap, and those excitations will
be unobservable at low energies. The interactions (or physics laws) that allow gapless
(or almost gapless) collective excitations to exist must be very special—and
'beautiful'.
If we believe that our vacuum can be viewed as a special many-body material,
then we have to conclude that there are no 'elementary' particles. All of the
so-called 'elementary' particles in our vacuum are actually low-energy collective
excitations and they may not be the building blocks of the fundamental theory.
The fundamental theory and its building blocks at high energies9 and short distances
are governed by a different set of physical laws. According to the point of
view of emergence, those laws may be specific, non-universal, and complicated.
The beautiful world and reasonable physical laws at low energies and long distances
emerge as a result of a 'natural selection': the physical laws that govern the
low-energy excitations should allow those excitations to exist at low energies. In a
sense, the 'natural selection' explains why our world is reasonable.
Someone who knows both condensed matter physics and high-energy physics
may object to the above picture because our vacuum appears to be very different
from the solids and liquids that we know of. For example, our vacuum contains
Dirac fermions (such as electrons and quarks) and gauge bosons (such as light),
while solids and liquids seemingly do not contain these excitations. It appears
that light and electrons are fundamental and cannot be emergent. So, to apply
the picture of emergence in many-body systems to elementary particles, we have
to address the following question: can gauge bosons and Dirac fermions emerge
from a many-body system? Or, more interestingly, can gauge bosons and Dirac
fermions emerge from a many-boson system?
The fundamental issue here is where do fermions and gauge bosons come
from? What is the origin of light and fermions? Can light and fermions be an
emergent phenomenon? We know that massless (or gapless) particles are very rare
in nature. If they exist, then they must exist for a reason. But what is the reason
behind the existence of the massless photons and nearly massless fermions (such
as electrons)? (The electron mass is smaller than the natural scale—the Planck
mass—by a factor of 1022 and can be regarded as zero for our purpose.) Can
many-body systems provide an answer to the above questions?
In the next few sections we will discuss some basic notions in many-body systems.
In particular, we will discuss the notion that leads to gapless excitations and
the notion that leads to emergent gauge bosons and fermions from local bosonic
models. We will see that massless photons and massless fermions can be emergent
phenomena.

1.3 Corner-stones of condensed matter physics
• Landau's symmetry-breaking theory (plus the renormalization group theory)
and Landau's Fermi liquid theory form the foundation of traditional
condensed matter physics.
The traditional many-body theory is based on two corner-stones, namely
Landau's Fermi liquid theory and Landau's symmetry-breaking theory (Landau,
1937; Ginzburg and Landau, 1950). The Fermi liquid theory is a perturbation
theory around a particular type of ground state—the states obtained by filling
single-particle energy levels. It describes metals, semiconductors, magnets, superconductors,
and superfluids. Landau's symmetry-breaking theory points out that
the reason that different phases are different is because they have different symmetries.
A phase transition is simply a transition that changes the symmetry. Landau's
symmetry-breaking theory describes almost all of the known phases, such as soh'd
phases, ferromagnetic and anti-ferromagnetic phases, superfluid phases, etc., and
all of the phase transitions between them.
Instead of the origin of light and fermions, let us first consider a simpler problem
of the origin of phonons. Using Landau's symmetry-breaking theory, we can
understand the origin of the gapless phonon. In Landau's symmetry-breaking theory,
a phase can have gapless excitations if the ground state of the system has a
special property called spontaneous breaking of the continuous symmetry (Nambu,
1960; Goldstone, 1961). Gapless phonons exist in a solid because the solid breaks
the continuous translation symmetries. There are precisely three kinds of gapless
phonons because the solid breaks three translation symmetries in the x, y, and z
directions. Thus, we can say that the origin of gapless phonons is the translational
symmetry breaking in solids.
It is quite interesting to see that our understanding of a gapless excitation—
phonon—is rooted in our understanding of the phases of matter. Knowing light to
be a massless excitation, one may perhaps wonder if light, just like a phonon,
is also a Nambu-Goldstone mode from a broken symmetry. However, experiments
tell us that a gauge boson, such as light, is really different from a
Nambu-Goldstone mode in 3 + 1 dimensions.
In the late 1970s, we felt that we understood, at least in principle, all of the
physics about phases and phase transitions. In Landau's symmetry-breaking theory,
if we start with a purely bosonic model, then the only way to get gapless
excitations is via spontaneous breaking of a continuous symmetry, which will
lead to gapless scalar bosonic excitations. It seems that there is no way to obtain
gapless gauge bosons and gapless fermions from symmetry breaking. This may
be the reason why people think that our vacuum (with massless gauge bosons
and nearly-gapless fermions) is very different from bosonic many-body systems
(which were believed to contain only gapless scalar bosonic collective excitations,
such as phonons). It seems that there does not exist any order that gives rise to
massless light and massless fermions. Due to this, we put light and fermions into a
different category to phonons. We regard them as elementary and introduce them
by hand into our theory of nature.
However, if we really believe that light and fermions, just like phonons, exist
for a reason, then such a reason must be a certain order in our vacuum that protects
their masslessness.10 Now the question is what kind of order can give rise to light
and fermions, and protect their masslessness? From this point of view, the very
existence of light and fermions indicates that our understanding of the states of
matter is incomplete. We should deepen and expand our understanding of the states
of matter. There should be new states of matter that contain new kinds of orders.
The new orders will produce light and fermions, and protect their masslessness.

1.4 Topological order and quantum order
• There is a new world beyond Landau's theories. The new world is rich and
exciting.
Our understanding of this new kind of order starts at an unexpected place—
fractional quantum Hall (FQH) systems. The FQH states discovered in 1982
(Tsui et aL, 1982; Laughlin, 1983) opened a new chapter in condensed mat
physics. What is really new in FQH states is that we have lost the two cornerstones
of the traditional many-body theory. Landau's Fermi liquid theory does
not apply to quantum Hall systems due to the strong interactions and correlations
in those systems. What is more striking is that FQH systems contain many different
phases at zero temperature which have the same symmetry. Thus, those
phases cannot be distinguished by symmetries and cannot be described by Landau's
symmetry-breaking theory. We suddenly find that we have nothing in the
traditional many-body theory that can be used to tackle the new problems. Thus,
theoretical progress in the field of strongly-correlated systems requires the introduction
of new mathematical techniques and physical concepts, which go beyond
the Fermi liquid theory and Landau's symmetry-breaking principle.
In the field of strongly-correlated systems, the developments in high-energy
particle theory and in condensed matter theory really feed upon each other. We
have seen a lot of field theory techniques, such as the nonlinear a-model, gauge
theory, bosonization, current algebra, etc., being introduced into the research of
strongly-correlated systems and random systems. This results in a very rapid development
of the field and new theories beyond the Fermi liquid theory and Landau's
symmetry-breaking theory. This book is an attempt to cover some of these new
developments in condensed matter theory.
One of the new developments is the introduction of quantum/topological order.
As FQH states cannot be described by Landau's symmetry-breaking theory, it was
proposed that FQH states contain a new kind of order—topological order (Wen,
1990, 1995). Topological order is new because it cannot be described by symmetry
breaking, long-range correlation, or local order parameters. None of the
usual tools that we used to characterize a phase apply to topological order. Despite
this, topological order is not an empty concept because it can be characterized
by a new set of tools, such as the number of degenerate ground states (Haldane
and Rezayi, 1985), quasiparticle statistics (Arovas et al, 1984), and edge states
(Halperin, 1982; Wen, 1992).
It was shown that the ground-state degeneracy of a topologically-ordered state
is robust against any perturbations (Wen and Niu, 1990). Thus, the ground-state
degeneracy is a universal property that can be used to characterize a phase. The
existence of topologically-degenerate ground states proves the existence of topological
order. Topological degeneracy can also be used to perform fault-tolerant
quantum computations (Kitaev, 2003).
The concept of topological order was recently generalized to quantum order
(Wen, 2002c) to describe new kinds of orders in gapless quantum states. One
way to understand quantum order is to see how it fits into a general classification
scheme of orders (see Fig. 1.1). First, different orders can be divided into
two classes: symmetry-breaking orders and non-symmetry-breaking orders. The
symmetry-breaking orders can be described by a local order parameter and can
be said to contain a condensation of point-like objects. The amplitude of the
condensation corresponds to the order parameter. All of the symmetry-breaking
orders can be understood in terms of Landau's symmetry-breaking theory. The
non-symmetry-breaking orders cannot be described by symmetry breaking, nor
by the related local order parameters and long-range correlations. Thus, they are
a new kind of order. If a quantum system (a state at zero temperature) contains
a non-symmetry-breaking order, then the system is said to contain a non-trivial
quantum order. We see that a quantum order is simply a non-symmetry-breaking
order in a quantum system.
Quantum orders can be further divided into many subclasses. If a quantum
state is gapped, then the corresponding quantum order will be called the topological
order. The low-energy effective theory of a topologically-ordered state will
be a topological field theory (Witten, 1989). The second class of quantum orders
appears in Fermi liquids (or free fermion systems). The different quantum orders
in Fermi liquids are classified by the Fermi su***ce topology (Lif***z, 1960). The
third class of quantum orders arises from a condensation of nets of strings (or simply
string-net condensation) (Wen, 2003a; Levin and Wen, 2003; Wen, 2003b).
This class of quantum orders shares some similarities with the symmetry-breaking
orders of 'particle' condensation.
We know that different symmetry-breaking orders can be classified by symmetry
groups. Using group theory, we can classify all of the 230 crystal orders
in three dimensions. The symmetry also produces and protects gapless collective
excitations—the Nambu-Goldstone bosons—above the symmetry-breaking
ground state. Similarly, different string-net condensations (and the corresponding
quantum orders) can be classified by mathematical object called projective symmetry
group (PSG) (Wen, 2002c). Using PSG, we can classify over 100 different
two-dimensional spin liquids that all have the same symmetry. Just like the symmetry
group, the PSG can also produce and protect gapless excitations. However,
unlike the symmetry group, the PSG produces and protects gapless gauge bosons
and fermions (Wen, 2002a,c; Wen and Zee, 2002). Because of this, we can say
that light and tnassless fermions can have a unified origin; they can emerge from
string-net condensations.
In light of the classification of the orders in Fig. 1.1, this book can be divided
into two parts. The first part (Chapters 3-5) deals with the symmetry-breaking
orders from 'particle' condensations. We develop the effective theory and study
the physical properties of the gapless Nambu-Goldstone modes from the fluctuations
of the order parameters. This part describes 'the origin of sound' and other
Nambu-Goldstone modes. It also describes the origin of the law of the 1/V4 dipolar
interaction between rotons in a superfluid. The second part (Chapters 7-10)
deals with the quantum/topological orders and string-net condensations. Again,
we develop the effective theory and study the physical properties of low-energy
collective modes. However, in this case, the collective modes come from the fluctuations
of condensed string-nets and give rise to gauge bosons and fermions. So,
the second part provides 'an origin of light and electrons', as well as other gauge
bosons and fermions. It also provides an origin of the 1/r2 Coulomb law (or, more
generally, the law of electromagnetism).

1.5 Origin of light and fermions
• The string-net condensation provides an answer to the origin of light and
fermions. It unifies gauge interactions and Fermi statistics.
We used to believe that, to have light and fermions in our theory, we have to
introduce by hand a fundamental U(l) gauge field and anti-commuting fermion
fields, because at that time we did not know of any collective modes that behave
like gauge bosons and fermions. However, due to the advances over the last twenty
years, we now know how to construct local bo sonic systems that have emergent
unconfined gauge bosons and/or fermions (Foerster et al, 1980; Kalmeyer and
Laughlin, 1987; Wen et al., 1989; Read and Sachdev, 1991; Wen, 1991a; Moessner
and Sondhi, 2001; Motrunich and Senthil, 2002; Wen, 2002a; Kitaev, 2003;
Levin and Wen, 2003). In particular, one can construct ugly bosonic spin models
on a cubic lattice whose low-energy effective theory is the beautiful quantum electrodynamics
(QED) and quantum chromodynamics (QCD) with emergent photons,
electrons, quarks, and gluons (Wen, 2003b).
This raises the following issue: do light and fermions in nature come from a
fundamental U(l) gauge field and anti-commuting fields as in the U(l) x SU(2) x
SU(3) standard model, or do they come from a particular quantum order in our
vacuum? Is Coulomb's law a fundamental law of nature or just an emergent phenomenon?
Clearly, it is more natural to assume that light and fermions, as well as
Coulomb's law, come from a quantum order in our vacuum. From the connections
between string-net condensation, quantum order, and massless gauge/fermion
excitations, we see that string-net condensation provides a way to unify light and
fermions. It is very tempting to propose the following possible answers to the three
fundamental questions about light and fermions.
What are light and fermions?
Light is the fluctuation of condensed string-nets (of arbitrary sizes). Fermions are
ends of condensed strings.
Where do light and fermions come from?
Light and fermions come from the collective motions of string-nets that fill the
space(see Fig. 1.2).
Why do light and fermions exist?
Light and fermions exist because our vacuum happens to have a property called
string-net condensation.
Had our vacuum chosen to have 'particle' condensation, then there would be
only Nambu-Goldstone bosons at low energies. Such a universe would be very
boring. String-net condensation and the resulting light and fermions provide a
much more interesting universe, at least interesting enough to support intelligent
life to study the origin of light and fermions.

1.6 Novelty is more important than correctness
• The Dao that can be stated cannot be eternal Dao. The Name that can be
named cannot be eternal Name. The Nameless is the origin of universe. The
Named is the mother of all matter.11
• What can be stated cannot be novel. What cannot be stated cannot be correct.
In this introduction (and in some parts of this book), I hope to give the reader
a sense of where we come from, where we stand, and where we are heading in
theoretical condensed matter physics. I am not trying to summarize the generally
accepted opinions here. Instead, I am trying to express my personal and purposely
exaggerated opinions on many fundamental issues in condensed matter physics
and high-energy physics. These opinions and pictures may not be correct, but I
hope they are stimulating. From our experience of the history of physics, we can
safely assume that none of the current physical theories are completely correct.
(According to Lao Zi, the theory that can be written down cannot be the eternal
theory, because it is limited by the mathematical symbols that we used to write
down the theory.) The problem is to determine in which way the current theories
are wrong and how to fix them. Here we need a lot of imagination and stimulation.

1.7 Remarks: evolution of the concept of elementary particles
• As time goes by, the status of elementary particles is downgraded from the
building blocks of everything to merely collective modes of, possibly, a lowly
bosonic model.
The Earth used to be regarded as the center of the universe. As times went by, its
status was reduced to merely one of the billions of planets in the universe. It appears that
the concept of elementary particles may have a similar fate.
At the beginning of human civilization, people realized that things can be divided into
smaller and smaller parts. Chinese philosophers theorized that the division could be continued
indefinitely, and hence that there were no elementary particles. Greek philosophers
assumed that the division could not be continued indefinitely. As a result, there exist ultimate
and indivisible particles—the building blocks of all matter. This may be the first concept of
elementary particles. Those ultimate particles were called atomos. A significant amount of
scientific research has been devoted to finding these atomos.
Around 1900, chemists discovered that all matter is formed from a few dozen different
kinds of particles. People jumped the gun and named them atoms. After the discovery of
the electron, people realized that elementary particles are smaller than atoms. Now, many
people believe that photons, electrons, quarks, and a few other particles are elementary
particles. Those particles are described by a field theory which is called the U( 1) x 577(2) x
577(3) standard model.
Although the £7(1) x 577(2) x SU(3) standard model is a very successful theory, now
most high-energy physicists believe that it is not the ultimate theory of everything. The
[/(I) x 577(2) x 577(3) standard model may be an effective theory that emerges from a
deeper structure. The question is from which structure may the standard model emerge?
One proposal is the grand unified theories in which the [/(I) x 577(2) x 577(3) gauge
group is promoted to 5(7(5) or even bigger gauge groups (Georgi and Glashow, 1974). The
grand unified theories group the particles in the (7(1) x SU(2) x 577(3) standard model
into very nice and much simpler structures. However, I would like to remark that I do not
regard the photon, electron, and other elementary particles to be emergent within the grand
unified theories. In the grand unified theories, the gauge structure and the Fermi statistics
were fundamental in the sense that the only way to have gauge bosons and fermions was to
introduce vector gauge fields and anti-commuting fermion fields. Thus, to have the photon,
electron, and other elementary particles, we had to introduce by hand the corresponding
gauge fields and fermion fields. Therefore, the gauge bosons and fermions were added by
hand into the grand unified theories; they did not emerge from a simpler structure.
The second proposal is the superstring theory (Green et a/., 1988; Polchinski, 1998).
Certain superstring models can lead to the effective (7(1) x SU(2) x 577(3) standard model
plus many additional (nearly) massless excitations. The gauge bosons and the graviton
are emergent because the superstring theory itself contains no gauge fields. However, the
Fermi statistics are not emergent. The electron and quarks come from the anti-commuting
fermion fields on a (1 + l)-dimensional world sheet. We see that, in the superstring theory,
the gauge bosons and the gauge structures are not fundamental, but the Fermi statistics
and the fermions are still fundamental.
Recently, people realized that there might be a third possibility—string-net condensation.
Banks et al. (1977) and Foerster ef a/. (1980) first pointed out that light can emerge as
low-energy collective modes of a local bosonic model. Levin and Wen (2003) pointed out
that even three-dimensional fermions can emerge from a local bosonic model as the ends
of condensed strings. Combining the two results, we find that the photon, electron, quark,
and gluon (or, more precisely, the QED and the QCD part of the 17(1) x SU(2) x 577(3) standard
model) can emerge from a local bosonic model (Wen, 2002a, 2003b) if the bosonic
model has a string-net condensation. This proposal is attractive because the gauge bosons
and fermions have a unified origin. In the string-net condensation picture, neither the
gauge structure nor the Fermi statistics are fundamental; all of the elementary particles
are emergent.
However, the third proposal also has a problem: we do not yet know how to produce the
SU(2) part of the standard model due to the chiral fermion problem. There are five deep
mysteries in nature, namely, identical particles, Fermi statistics, gauge structure, chiral
fermions, and gravity. The string-net condensation only provides an answer to the first three
mysteries; there are two more to go.

2
PATH INTEGRAL FORMULATION OF QUANTUM
MECHANICS
This book is about the quantum behavior of many-body systems. However, the
standard formulation of the quantum theory in terms of wave functions and the
Schrodinger equation is not suitable for many-body systems. In this chapter, we
introduce a semiclassical picture and path integral formalism for quantum theory.
The path integral formalism can be easily applied to many-body systems. Here,
we will use one-particle systems as concrete examples to develop the formalism.
We will also apply a path integral formalism to study quantum friction, simple
quantum circuits, etc.

2.1 Semiclassical picture and path integral
• A semiclassical picture and a path integral formulation allow us to visualize
quantum behavior. They give us a global view of a quantum system.
When we are thinking of a physical problem or trying to understand a phenomenon,
it is very important to have a picture in our mind to mentally visualize
the connection between different pieces of a puzzle. Mental visualization is easier
when we consider a classical system because the picture of a classical system is
quite close to what we actually see in our everyday life. However, when we consider
a quantum system, visualization is much harder. This is because the quantum
world does not resemble what we see every day. In the classical world, we see
various objects. With a little abstraction, we view these objects as collections
of particles. The concept of a particle is the most important concept in classical
physics. It is so simple and plain that people take it for granted and do not bother
to formulate a physical law to state such an obvious truth. However, it is this obvious
truth which turns out to be false in the quantum world. The concept of a particle
with a position and velocity simply does not exist in the quantum world. So, the
challenge is how to visualize anything in the quantum world where the concept of
a particle (i.e. the building block of objects) does not exist.
The path integral formalism is an attempt to use a picture of the classical world
to describe the quantum world. In other words, it is a bridge between the classical
world (where we have our experiences and pictures) and the quantum world (which
represents realities). The path integral formalism will help us to visualize quantum
behavior in terms of the pictures of corresponding classical systems.

2.1.1 Propagator of a particle
• Concept of a propagator.
• The pole structure of a propagator in ui space.

2.1.2 Path integral representation of the propagator
• The path integral is a particular representation of the law of superposition in
quantum physics.
The spirit of the path integral formulation of a quantum system is very simple.
Consider that a particle propagates from position lxa} to position |xf,) via an intermediate
state 2.1.2 Path integral representation of the propagator
• The path integral is a particular representation of the law of superposition in
quantum physics.
The spirit of the path integral formulation of a quantum system is very simple.
Consider that a particle propagates from position lxa} to position |xf,) via an intermediate
state ln), n = 1,2,.... Let the amplitude from |.xa) to lXb) via state ln)
be given by An; then the total amplitude from lxa} to lxb} is J]n An. Now imagine
that we divide the propagation from lxa} to lxb) into many time slices. Each
time slice has its own set of intermediate states. The propagation from lxa) to lxb)
can be viewed as the sum of the paths that go through the intermediate states on
each time slice. This leads to a picture of the path integral representation of the
propagation amplitude.

Path integral formulation

From Wikipedia, the free encyclopedia



[ltr]This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see line integral.[/ltr]


[th]Quantum mechanics[/th]

Uncertainty principle

Introduction Glossary History

Background[show]

Fundamentals[show]

Experiments[show]

Formulations[show]

Equations[show]

Interpretations[show]

Advanced topics[show]

Scientists[show]

v t e

[ltr]
The path integral formulation ofquantum mechanics is a description of quantum theory which generalizes the action principle ofclassical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude.
The basic idea of the path integral formulation can be traced back toNorbert Wiener, who introduced theWiener integral for solving problems in diffusion and Brownian motion.^[1] This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 paper.^[2] The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier, in the course of hisdoctoral thesis work with John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than aHamiltonian) as a starting point.
This formulation has proven crucial to the subsequent development oftheoretical physics, because it is manifestly symmetric between time and space. Unlike previous methods, the path-integral allows a physicist to easily change coordinates between very different canonical descriptions of the same quantum system.
The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unifiedquantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is adiffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possiblerandom walks. For this reason path integrals were used in the study ofBrownian motion and diffusion a while before they were introduced in quantum mechanics.^[3][/ltr]

• 1 Quantum action principle
  
• 2 Feynman's interpretation
  
• 3 Concrete formulation
  
  
  
  • 3.1 Time-slicing definition
    
  • 3.2 Free particle
    
  • 3.3 The Schrödinger equation
    
  • 3.4 Equations of motion
    
  • 3.5 Stationary phase approximation
    
  • 3.6 Canonical commutation relations
    
  • 3.7 Particle in curved space
    
  • 3.8 The path integral and the partition function
    
  • 3.9 Measure theoretic factors
    
  
  
  
• 4 Quantum field theory
  
  
  
  • 4.1 The propagator
    
  • 4.2 Functionals of fields
    
  • 4.3 Expectation values
    
  • 4.4 As a probability
    
  • 4.5 Schwinger–Dyson equations
    
  
  
  
• 5 Localization
  
• 6 Functional identity
  
  
  
  • 6.1 Ward–Takahashi identities
    
  
  
  
• 7 The need for regulators and renormalization
  
• 8 The path integral in quantum-mechanical interpretation
  
• 9 Quantum Gravity
  
• 10 See also
  
• 11 References
  
• 12 Notes
  
• 13 Suggested reading
  
• 14 External links
  

[ltr]

Quantum action principle[edit]
In quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time-translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative imaginary unit, −i). For states with a definite energy, this is a statement of the De Broglie relationbetween frequency and energy, and the general relation is consistent with that plus the superposition principle.
But the Hamiltonian in classical mechanics is derived from a Lagrangian, which is a more fundamental quantity relative to special relativity. The Hamiltonian tells you how to march forward in time, but the time is different in different reference frames. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.
The Hamiltonian is a function of the position and momentum at one time, and it tells you the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a Legendre transform, and the condition that determines the classical equations of motion (the Euler–Lagrange equations) is that the action is a minimum.
In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. So what does the Legendre transform mean? In classical mechanics, with discretization in time,

and

where the partial derivative with respect to q holds q(t + ε) fixed. The inverse Legendre transform is:

where

and the partial derivative now is with respect to p at fixed q.
In quantum mechanics, the state is a superposition of different states with different values of q, or different values of p, and the quantities p and qcan be interpreted as noncommuting operators. The operator p is only definite on states that are indefinite with respect to q. So consider two states separated in time and act with the operator corresponding to the Lagrangian:

If the multiplications implicit in this formula are reinterpreted as matrix multiplications, what does this mean?
It can be given a meaning as follows: The first factor is

If this is interpreted as doing a matrix multiplication, the sum over all states integrates over all q(t), and so it takes the Fourier transform in q(t), to change basis to p(t). That is the action on the Hilbert space – change basis to p at time t.
Next comes:

or evolve an infinitesimal time into the future.
Finally, the last factor in this interpretation is

which means change basis back to q at a later time.
This is not very different from just ordinary time evolution: the H factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just doing Fourier transforms to change to a pure q basis from an intermediate p basis.
Another way of saying this is that since the Hamiltonian is naturally a function of p and q, exponentiating this quantity and changing basis from pto q at each step allows the matrix element of H to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to Paul Dirac.[/ltr]



[ltr]
Dirac further noted that one could square the time-evolution operator in the S representation

and this gives the time evolution operator between time t and time t + 2ε. While in the H representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the S representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of q(0) and the later one with a fixed value of q(t). The result is a sum over paths with a phase which is the quantum action. Crucially, Dirac identified in this paper the deep quantum mechanical reason for the principle of least action controlling the classical limit (see quote box).
Feynman's interpretation[edit]
Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the canonical commutation relations from this rule. This was done by Feynman.^[4]
Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics from the following postulates:[/ltr]

1. The probability for an event is given by the modulus length squared of a complex number called the "probability amplitude".
   
2. The probability amplitude is given by adding together the contributions of all paths in configuration space.
   
3. The contribution of a path is proportional to . while S is theaction given by the time integral of the Lagrangian along the path.
   

[ltr]
In order to find the overall probability amplitude for a given process, then, one adds up, or integrates, the amplitude of postulate 3 over the space ofall possible paths of the system in between the initial and final states, including those that are absurd by classical standards. In calculating the amplitude for a single particle to go from one place to another in a given time, it is correct to include paths in which the particle describes elaboratecurlicues, curves in which the particle shoots off into outer space and flies back again, and so forth. The path integral assigns to all these amplitudes equal weight but varying phase, or argument of the complex number. Contributions from paths wildly different from the classical trajectory may be suppressed by interference (see below).
Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics when the Hamiltonian is quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for theHamiltonian corresponding to the given action.
Concrete formulation[edit]
Feynman's postulates can be interpreted as follows:
Time-slicing definition[edit]
For a particle in a smooth potential, the path integral is approximated byzig-zag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position x_a at time t_a to x_b at time t_b, the time sequence

can be divided up into n + 1 little segments t_j − t_{j − 1}, where j = 1,...,n + 1, of fixed duration

This process is called time-slicing.
An approximation for the path integral can be computed as proportional to

where  is the Lagrangian of the 1d system with position variable x(t) and velocity v = ẋ(t) considered (see below), and dx_jcorresponds to the position at the jth time step, if the time integral is approximated by a sum of n terms.^{[note 1]}
In the limit n → ∞, this becomes a functional integral, which – apart from a nonessential factor – is directly the product of the probability amplitudes  – more precisely, since one must work with a continuous spectrum, the respective densities – to find the quantum mechanical particle at t_a in the initial state x_a and at t_b in the final state x_b.
Actually  is the classical Lagrangian of the one-dimensional system considered, also

where  is the Hamiltonian,
, and the above-mentioned "zigzagging" corresponds to the appearance of the terms:
In the Riemannian sum approximating the time integral, which are finally integrated over x₁ to x_n with the integration measure dx₁...dx_nx̃_j is an arbitrary value of the interval corresponding to j, e.g. its center, (x_j + x_{j − 1})/2.
Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.[/ltr]


[ltr]
Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the Coulomb potential e²/r at the origin. Only after replacing the timet by another path-dependent pseudo-time parameter

the singularity is removed and a time-sliced approximation exists, that is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by İsmail Hakkı Duru and Hagen Kleinert.^[5][6] The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the Duru–Kleinert transformation.
Free particle[edit]
The path integral representation gives the quantum amplitude to go from point x to point y as an integral over all paths. For a free particle action (m= 1, ħ = 1):

the integral can be evaluated explicitly.
To do this, it is convenient to start without the factor i in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions.

Splitting the integral into time slices:

where the Dx is interpreted as a finite collection of integrations at each integer multiple of ε. Each factor in the product is a Gaussian as a function of x(t + ε) centered at x(t) with variance ε. The multiple integrals are a repeated convolution of this Gaussian G_ε with copies of itself at adjacent times.

Where the number of convolutions is T/ε. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications.

The Fourier transform of the Gaussian G is another Gaussian of reciprocal variance:

and the result is:

The Fourier transform gives K, and it is a Gaussian again with reciprocal variance:

The proportionality constant is not really determined by the time slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time-slices the time-evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process.
The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the central limit theorem, which can be interpreted as the first historical evaluation of a statistical path integral.
The probability interpretation gives a natural normalization choice. The path integral should be defined so that:

This condition normalizes the Gaussian, and produces a Kernel which obeys the diffusion equation:

For oscillatory path integrals, ones with an i in the numerator, the time-slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to the time increment . This is closely related to Wick rotation. Then the same convolution argument as before gives the propagation kernel:

Which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation

This means that any superposition of K's will also obey the same equation, by linearity. Defining

then ψ_t obeys the free Schrödinger equation just as K does:

The Schrödinger equation[edit]
Main article: Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.

Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of ẋ, the path integral has most weight fory close to x. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. The exponential of the action is

The first term rotates the phase of ψ(x) locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to i times a diffusion process. To lowest order in ε they are additive; in any case one has with (1):

As mentioned, the spread in ψ is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase which slowly varies from point to point from the potential:

and this is the Schrödinger equation. Note that the normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
Equations of motion[edit]
Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of x and ẋvariables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.
Start by considering the path integral with some fixed initial state

Now note that x(t) at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting:  where ε(t) is a different shift at each time but ε(0) = ε(T) = 0, since the endpoints are not integrated:

The change in the integral from the shift is, to first infinitesimal order in epsilon:

which, integrating by parts in t, gives:

But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of ε(t). The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:

this is the Heisenberg equations of motion.
If the action contains terms which multiply ẋ and x, at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism.
Stationary phase approximation[edit]
If the variation in the action exceeds ħ by many orders of magnitude, we typically have destructive phase interference other than in the vicinity of those trajectories satisfying the Euler–Lagrange equation, which is now reinterpreted as the condition for constructive phase interference.
Canonical commutation relations[edit]
The formulation of the path integral does not make it clear at first sight that the quantities x and p do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still there.^[7]
To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by i:

The quantity x(t) is fluctuating, and the derivative is defined as the limit of a discrete difference.

Note that the distance that a random walk moves is proportional to √t, so that:

This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.
The quantity x ẋ is ambiguous, with two possible meanings:

In elementary calculus, the two are only different by an amount which goes to zero as ε goes to zero. But in this case, the difference between the two is not zero:

give a name to the value of the difference for any one random walk:

and note that f(t) is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian

and the equations of motion for f derived from extremizing the action Scorresponding to  just set it equal to 1. In physics, such a quantity is "equal to 1 as an identity operator". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose.
Defining the time order to be the operator order:

This is called the Itō lemma in stochastic calculus, and the (euclideanized) canonical commutation relations in physics.
For a general statistical action, a similar argument shows that

and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation,

Particle in curved space[edit]
For a particle in curved space the kinetic term depends on the position and the above time slicing cannot be applied, this being a manifestation of the notorious operator ordering problem in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation (nonholonomic mapping explained here).
The path integral and the partition function[edit]
The path integral is just the generalization of the integral above to all quantum mechanical problems—
  where  
is the action of the classical problem in which one investigates the path starting at time t=0 and ending at time t = T, and Dx denotes integration over all paths. In the classical limit, , the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.^[8]
The connection with statistical mechanics follows. Considering only paths which begin and end in the same configuration, perform the Wick rotation t→ it, i.e., make time imaginary, and integrate over all possible beginning/ending configurations. The path integral now resembles thepartition function of statistical mechanics defined in a canonical ensemblewith temperature . Strictly speaking, though, this is the partition function for a statistical field theory.
Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by

where the state α is evolved from time t = 0. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time iT is given by

which is precisely the partition function of statistical mechanics for the same system at temperature quoted earlier. One aspect of this equivalence was also known to Schrödinger who remarked that the equation named after him looked like the diffusion equation after Wick rotation.[/ltr]

Measure theoretic factors[edit]

Sometimes (e.g. a particle moving in curved space) we also have measure-theoretic factors in the functional integral.

This factor is needed to restore unitarity.

For instance, if
,
then it means that each spatial slice is multiplied by the measure √g. This measure can't be expressed as a functional multiplying the  measure because they belong to entirely different classes.

Quantum field theory[edit]

[th]Quantum field theory[/th]

Feynman diagram

History

Background[show]

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Incomplete theories[show]

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v t e

The path integral formulation was very important for the development of quantum field theory. Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time, and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation

for x and y two simultaneous spatial positions, and this is not a relativistically invariant concept. The results of a calculation are covariant at the end of the day, but the symmetry is not apparent in intermediate stages. If *** field theory calculations did not produce infinite answers in the continuum limit, this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a careful limiting procedure.

The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg type operator algebra to operator product rules which are new relations difficult to see in the old formalism.

Further, different choices of canonical variables lead to very different seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons, the Feynman path integral has made earlier formalisms largely obsolete.

The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation. The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful mathematics not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly. This required physicists to invent an entirely new mathematical object – the Grassmann variable – which also allowed changes of variables to be done naturally, as well as allowingconstrained quantization.

The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some *** identities fail.

The propagator[edit]

In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.

The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point x to point y in time T.

This is called the propagator. Superposing different values of the initial position  with an arbitrary initial state  constructs the final state.

For a spatially homogeneous system, where K(x, y) is only a function of (x − y), the integral is a convolution, the final state is the initial state convolved with the propagator.

For a free particle of mass m, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time and the solution must be a normalized Gaussian:

Taking the Fourier transform in (x − y) produces another Gaussian:

and in p-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending K(p; T) to be zero for negative times, gives the Green's Function, or the frequency space propagator:

Which is the reciprocal of the operator which annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the p-space representation.

The infinitesimal term in the denominator is a small positive number which guarantees that the inverse Fourier transform in E will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of E where there is no singularity. This guarantees that K propagates the particle into the future and is the reason for the subscript on G. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.

It is also possible to reexpress the nonrelativistic time evolution in terms of propagators which go toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the gaussian t is replaced by (−t). In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction.

Given the nearly identical only change is the sign of E and ε. The parameter E in the Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past.

For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths which travel between two points in a fixed proper time, as measured along the path. These paths describe the trajectory of a particle in space and in time.

The integral above is not trivial to interpret, because of the square root. Fortunately, there is a heuristic trick. The sum is over the relativistic arclength of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary time. The function  can be evaluated when the sum is over paths in Euclidean space.

This describes a sum over all paths of length  of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to , and each step is less likely the longer it is. By the central limit theorem, the result of many independent steps is a Gaussian of variance proportional to .

The usual definition of the relativistic propagator only asks for the amplitude is to travel from x to y, after summing over all the possible proper times it could take.

Where  is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor, and can be absorbed into the constant α.

This is the Schwinger representation. Taking a Fourier transform over the variable (x − y) can be done for each value of  separately, and because each separate  contribution is a Gaussian, gives whose Fourier transform is another Gaussian with reciprocal width. So in p-space, the propagator can be reexpressed simply:

Which is the Euclidean propagator for a scalar particle. Rotating p₀ to be imaginary gives the usual relativistic propagator, up to a factor of (−i) and an ambiguity which will be clarified below.

This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by partial fractions:

For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near p₀ = m. When convolving with the propagator, which in p space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near p₀ = m, the dominant first term has the form:

This is the expression for the nonrelativistic Green's function of a free Schrödinger particle.

The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies which are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy.

The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where t → −∞ of the first term must vanish, while the t → +∞ limit of the second term must vanish. In the Fourier transform, this means shifting the pole in p₀ slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:

Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of p₀. The terms can be recombined:

Which when factored, produces opposite sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The ε term introduces a small imaginary part to the α = m², which in the Minkowski version is a small exponential suppression of long paths.

So in the relativistic case, the Feynman path-integral representation of the propagator includes paths which go backwards in time, which describe antiparticles. The paths which contribute to the relativistic propagator go forward and backwards in time, and the interpretation of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.

Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses which are nonzero outside the lightcone, meaning that it is impossible to keep a particle from travelling faster than light. Such a particle cannot have a Greens function which is only nonzero in the future in a relativistically invariant theory.

Functionals of fields[edit]

However, the path integral formulation is also extremely important in directapplication to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space. The action is referred to technically as a functional of the field: S[ϕ] where the field ϕ(x^μ) is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value. In principle, one integrates Feynman's amplitude over the class of all possible combinations of values that the field could have anywhere inspace–time.

Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward *** these functional integrals mathematically precise.

Such a functional integral is extremely similar to the partition function instatistical mechanics. Indeed, it is sometimes called a partition function, and the two are essentially mathematically identical except for the factor ofi in the exponent in Feynman's postulate 3. Analytically continuing the integral to an imaginary time variable (called a Wick rotation) makes the functional integral even more like a statistical partition function, and also tames some of the mathematical difficulties of working with these integrals.

Expectation values[edit]

In quantum field theory, if the action is given by the functional  of field configurations (which only depends locally on the fields), then the time ordered vacuum expectation value of polynomially bounded functional F, <F>, is given by

The symbol  here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space–time. As stated above, we put the unadorned path integral in the denominator to normalize everything properly.

As a probability[edit]

Strictly speaking the only question that can be asked in physics is: "What fraction of states satisfying condition A also satisfy condition B?" The answer to this is a number between 0 and 1 which can be interpreted as aprobability which is written as P(B|A). In terms of path integration, since  this means:

where the functional O_in[ϕ] is the superposition of all incoming states that could lead to the states we are interested in. In particular this could be a state corresponding to the state of the Universe just after the big bangalthough for actual calculation this can be simplified using heuristic methods. Since this expression is a quotient of path integrals it is naturally normalised.

Schwinger–Dyson equations[edit]

Main article: Schwinger–Dyson equation

Since this formulation of quantum mechanics is analogous to classical action principles, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.

In the language of functional analysis, we can write the Euler–Lagrange equations as

(the left-hand side is a functional derivative; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the Schwinger–Dyson equations.

If the functional measure  turns out to be translationally invariant (we'll assume this for the rest of this article, although this does not hold for, let's say nonlinear sigma models) and if we assume that after a Wick rotation

which now becomes

for some H, goes to zero faster than a reciprocal of any polynomial for large values of φ, we can integrate by parts (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation:

for any polynomially bounded functional F.

in the deWitt notation.

These equations are the analog of the on shell EL equations.

If J (called the source field) is an element of the dual space of the field configurations (which has at least an affine structure because of the assumption of the translational invariance for the functional measure), then, the generating functional Z of the source fields is defined to be:

Note that

or

where

Basically, if  is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of QFT, unlike its Wick rotatedstatistical mechanics analogue, because we have time orderingcomplications here!), then  are its moments and Z is its Fourier transform.

If F is a functional of φ, then for an operator K, F[K] is defined to be the operator which substitutes K for φ. For example, if

and G is a functional of J, then

Then, from the properties of the functional integrals

we get the "master" Schwinger–Dyson equation:

or

If the functional measure is not translationally invariant, it might be possible to express it as the product  where M is a functional and is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the target space is diffeomorphic to Rⁿ. However, if the target manifold is some topologically nontrivial space, the concept of a translation does not even make any sense.

In that case, we would have to replace the  in this equation by another functional 

If we expand this equation as a Taylor series about J = 0, we get the entire set of Schwinger–Dyson equations.

Localization[edit]

The path integrals are usually thought of as being the sum of all paths through an infinite space–time. However, in Local quantum field theory we would restrict everything to lie within a finite causally complete region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory.

Functional identity[edit]

If we perform a Wick rotation inside the functional integral, professors J. Garcia and Gerard 't Hooft showed using a functional differential equation, that



where S is the Wick-rotated classical action of the particle, J is the classical action with an extra term "x", delta (here) is the functional derivative operator and

Ward–Takahashi identities[edit]
See main article Ward–Takahashi identity.
Now how about the on shell Noether's theorem for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.

Let's just assume for simplicity here that the symmetry in question is local (not local in the sense of a gauge symmetry, but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a Lagrangian, and that  for some function f where f only depends locally on φ (and possibly the spacetime position).

If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless f=0 or something. Here, Q is a derivation which generates the one parameter group in question. We could have antiderivations as well, such as BRSTand supersymmetry.

Let's also assume  for any polynomially bounded functional F. This property is called the invariance of the measure. And this does not hold in general. See anomaly (physics) for more details.

Then,

which implies

where the integral is over the boundary. This is the quantum analog of Noether's theorem.

Now, let's assume even further that Q is a local integral

where

so that

where

(this is assuming the Lagrangian only depends on φ and its first partial derivatives! More general Lagrangians would require a modification to this definition!). Note that we're NOT insisting that q(x) is the generator of a symmetry (i.e. we are not insisting upon the gauge principle), but just thatQ is. And we also assume the even stronger assumption that the functional measure is locally invariant:

Then, we would have

Alternatively,

The above two equations are the Ward–Takahashi identities.

Now for the case where f=0, we can forget about all the boundary conditions and locality assumptions. We'd simply have

Alternatively,

The need for regulators and renormalization[edit]

Path integrals as they are defined here require the introduction ofregulators. Changing the scale of the regulator leads to therenormalization group. In fact, renormalization is the major obstruction to *** path integrals well-defined.

The path integral in quantum-mechanical interpretation[edit]

In one philosophical interpretation of quantum mechanics, the "sum over histories" interpretation, the path integral is taken to be fundamental and reality is viewed as a single indistinguishable "class" of paths which all share the same events. For this interpretation, it is crucial to understand what exactly an event is. The sum over histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin (see the reference below) claim the interpretation explains the Einstein–Podolsky–Rosen paradox without resorting to nonlocality. (Note that the Copenhagen/pragmatism interpretation claims there is no paradox—only a sloppy materialism motivated question on the part of EPR—Joseph Wienberg a lecture. On the other hand, the fact that the EPR thought experiment (and its result) does represent the results of a QM experiment says that (despite the path dependence of parallelness/anti-parallelness in curved space) all contributions of paths close to black holes cancel in the action for an EPR style experiment here on earth.)

Some advocates of interpretations of quantum mechanics emphasizingdecoherence have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.

Quantum Gravity[edit]

Whereas in Quantum Theory the path integral formulation is fully equivalent to other formulations, it may be that it can be extended to quantum gravity, which would make it different from the Hilbert spacemodel. Feynman had some success in this direction and his work has been extended by Hawking and others.^[9] Approaches that use this method include Causal Dynamical Triangulations, Tensor models and spinfoams.

See also[edit]

• Theoretical and experimental justification for the Schrödinger equation
  
  
• Static forces and virtual-particle exchange
  
  
• Feynman checkerboard
  
  
• Propagators
  
  
• Wheeler–Feynman absorber theory
  
  
• Feynman–Kac formula
  
  

[ltr]References[size=13][edit]
[/ltr][/size]

1. Jump up^ Masud Chaichian, Andrei Pavlovich Demichev (2001). "Introduction".Path Integrals in Physics Volume 1: Stochastic Process & Quantum Mechanics. Taylor & Francis. p. 1 ff. ISBN 0-7503-0801-X.
   
2. Jump up^ Dirac, Paul A. M. (1933). "The Lagrangian in Quantum Mechanics".Physikalische Zeitschrift der Sowjetunion 3: 64–72.; also see Van Vleck, John H (1928). "The correspondence principle in the statistical interpretation of quantum mechanics". Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sciences) 14 (2): 178188.
   
3. Jump up^ Kleinert, H. (1989). "6". Gauge Fields in Condensed Matter. 1: Superflow and Vortex Lines. Singapore: World Scientific. ISBN 9971-5-0210-0.
   
4. Jump up^ Both noted that in the limit of action that is large compared to the reduced Planck's constant ħ (using natural units, ħ = 1), the path integral is dominated by solutions which are in the neighbourhood of stationary points of the action.
   
5. Jump up^ Duru, H; Hagen Kleinert (1979-06-18). "Solution of the path integral for the H-atom". Physics Letters 84B (2). Retrieved 2007-11-25.
   
6. Jump up^ For details see Chapter 13 in Kleinert's book cited above.
   
7. Jump up^ Feynman, R. P. (1948). "Space-Time Approach to Non-Relativistic Quantum Mechanics". "Reviews of Modern Physics" 20 (2): 367–387.Bibcode:1948RvMP...20..367F. doi:10.1103/RevModPhys.20.367.
   
8. Jump up^ Feynman, Richard P. (Richard Phillips); Hibbs, Albert R.; Styer, Daniel F. (2010). Quantum Mechanics and Path Integrals. Mineola, N.Y.: Dover Publications. pp. 29–31. ISBN 0-486-47722-3.
   
9. Jump up^ "Most of the Good Stuff", Memories Of Richard Feynman, edited by Laurie M. Brown and John S. Rigden, American Institute of Physics, the chapter by Murray Gell-Mann.
   

[size][ltr]
Notes[edit]
[/ltr][/size]

1. Jump up^ For a simplified, step by step, derivation of the above relation see Path Integrals in Quantum Theories: A Pedagogic 1st Step pdf vers
   

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Suggested reading[edit]
[/ltr][/size]

• Feynman, R. P. and Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals. New York: McGraw-Hill. ISBN 0-07-020650-3. The historical reference, written by the inventor of the path integral formulation himself and one of his students.
  
• Hagen Kleinert (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (4th ed.). Singapore: World Scientific. ISBN 981-238-107-4.
  
• Zinn Justin, Jean (2004). Path Integrals in Quantum Mechanics. Oxford University Press. ISBN 0-19-856674-3. A highly readable introduction to the subject.
  
• Schulman, Larry S. (1981). Techniques & Applications of Path Integration. New York: John Wiley & Sons. ISBN 0486445283. A modern reference on the subject.
  
• Ahmad, Ishfaq (1971). Mathematical Integrals in Quantum Nature. The Nucleus. pp. 189–209.
  
• Grosche, Christian & Steiner, Frank (1998). Handbook of Feynman Path Integrals. Springer Tracts in Modern Physics 145. Springer-Verlag. ISBN 3-540-57135-3.
  
• Ryder, Lewis H. (1985). Quantum Field Theory. Cambridge University Press. ISBN 0-521-33859-X. Highly readable textbook; introduction to relativistic QFT for particle physics.
  
• Rivers, R.J. (1987). Path Integrals Methods in Quantum Field Theory. Cambridge University Press. ISBN 0-521-25979-7.
  
• Albeverio, S. & Hoegh-Krohn. R. (1976). Mathematical Theory of Feynman Path Integral. Lecture Notes in Mathematics 523. Springer-Verlag. ISBN 0-387-07785-5.
  
• Glimm, James, and Jaffe, Arthur (1981). Quantum Physics: A Functional Integral Point of View. New York: Springer-Verlag. ISBN 0-387-90562-6.
  
• Gerald W. Johnson and Michel L. Lapidus (2002). The Feynman Integral and Feynman's Operational Calculus. Oxford Mathematical Monographs. Oxford University Press. ISBN 0-19-851572-3.
  
• Etingof, Pavel (2002). "Geometry and Quantum Field Theory". MIT OpenCourseWare. This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.
  
• Zee, Anthony. Quantum Field Theory in a Nutshell (Second ed.). Princeton University Press. ISBN 978-0-691-14034-6. A great introduction to Path Integrals (Chapter 1) and QFT in general.
  
• Grosche, Christian (1992). "An Introduction into the Feynman Path Integral". arXiv:hep-th/9302097.
  
• MacKenzie, Richard (2000). "Path Integral Methods and Applications".arXiv:quant-ph/0004090.
  
• DeWitt-Morette, Cécile (1972). "Feynman's path integral: Definition without limiting procedure". Communication in Mathematical Physics 28(1): 47–67. Bibcode:1972CMaPh..28...47D. MR 0309456.doi:10.1007/BF02099371.
  
• Sinha, Sukanya; Sorkin, Rafael D. (1991). "A Sum-over-histories Account of an EPR(B) Experiment". Found. of Phys. Lett. 4 (4): 303–335. Bibcode:1991FoPhL...4..303S. doi:10.1007/BF00665892.
  
• Cartier, Pierre; DeWitt-Morette, Cécile (1995). "A new perspective on Functional Integration". Journal of Mathematical Physics 36 (5): 2137–2340. Bibcode:1995JMP....36.2237C. arXiv:funct-an/9602005.doi:10.1063/1.531039.
  

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• Path integral on Scholarpedia
  
• Path Integrals in Quantum Theories: A Pedagogic 1st Step
  

String–Net Models with ZN Fusion algebra
Ling-Yan Hung∗
Perimeter Institute for Theoretical Physics
31 Caroline St N, Waterloo, ON N2L 2Y5, Canada
Yidun Wan†
Research Center for Quantum computation, Kinki University
Kowakae 3-4-1, Higashi-Osaka 577-8502, Osaka, Japan
(Dated: February 3, 2015)
We study the Levin–Wen string–net model with a ZN type fusion algebra. Solutions of the local
constraints of this model correspond to ZN gauge theory and double Chern–Simons theories with
quantum groups. For the first time, we explicitly construct a spin-(N − 1)/2 model with ZN gauge
symmetry on a triangular lattice as an exact dual model of the string–net model with a ZN type
fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled
to a ZN gauge field living on the links of the triangular lattice. The ungauged ZN lattice spin
models are a class of quantum systems that bear symmetry-protected topological phases that may
be classified by the third cohomology group H3(ZN,U(1)) of ZN. Our results apply also to any case
where the fusion algebra is identified with a finite group algebra or a quantum group algebra.

The classification of possible phases of matter is central
to the study of condensed matter physics. Until very
recently different phases of matter have been associated
with symmetry breaking, which can be very succinctly
described in the paradigm of Landau’s effective theory.
Important leaps in our understanding come about when
it is realized that new phases of matter can arise even as
no symmetry breaking is involved.
It is therefore very important to give a systematic survey
of these states of matter, and ideally, provide a complete
classification of them.
Very broadly speaking, gapped quantum phases of
matter can be divided into two classes: namely those involving
long range entanglement (LRE) and those involving
only short range entanglement (SRE). When symmetries
are present, SRE displays a myriad of phases. For
example Landau’s paradigm of spontaneous symmetry
breaking belongs to the class of SRE. When symmetries
are unbroken, there are also distinct phases of matter,
often termed the symmetry protected topological (SPT)
phases. Their classification in terms of group cohomology
is recently given in Ref1.
On the other hand, the LRE phases of matter are examples
that realize topological order, in which they display
features such as robust ground state degeneracies,
non-Abelian statistics of quasi–particle excitations, and
in many cases protected edge excitations. The classic examples
of these phases include the (fractional) quantum
Hall states and chiral spin liquids. There is a very general
framework supplying exactly solvable models that
incorporates a large class of LRE phases, notably those
preserving time-reversal symmetry. This is called the
string–net models2, and it has been known that the tensor
category theory is the mathematical framework that
underlies these models.
Very recently, a connection is discovered between a
specific SPT phase, namely an Ising spin model with
Z2 symmetry, and a LRE phase described by a string
net model3,4. In particular in the construction in4 it is
found that when the Z2 symmetry of the spin model is
gauged, it admits a dual description in terms of a string
net model whose fusion rules are given exactly by Z2.
It was conjectured that for a general SPT phase with
discrete symmetry G, by gauging G it admits a string
net dual description with fusion rules also given by the
product rule of G.
In this paper, by studying the explicit examples of
string–net models with ZN type fusion algebra, we construct
such a map between the string net models and
the gauged SPT model. Although our construction is
based on ZN fusion algebra, it is immediately applicable
to more general discrete groups G. This implies that
the classification of SPT phases provided by group cohomology
in 2+1 dimensions via H3(G,U(1)) described in
Ref1 indirectly provide classifications of the corresponding
string net models. We support this claim also by
studying the rescaling redundancy of the 6j symbols that
characterize a given string net model. We find that when
the fusion rules coincide with the product rule of a group
G, the 6j symbols can be interpreted as a 3-cocycle and
that their rescaling redundancy can be understood as an
equivalence between these 3-cocycles up to a co-boundary
in the context of group cohomology19. Therefore these
6j symbols admit a classification by H3(G,U(1)), coinciding
with that of the dual SPT phases.
Our paper is organized as follows. We begin in section
II with a review of the basic ingredients of the string
net models. In section III we revisit the rescaling redundancy
of the 6j symbols and point out its relationship
with group cohomology. In section IV, we study string–
net models with ZN type fusion algebra in greater details,
and collect a number of useful facts about them. Some
further details and the explicit forms of 6j symbols corresponding
to ZN fusion algebra of various N are relegated
to the appendix. In section V, we construct the explicit
map between the string–net models with ZN fusion algebra
and the corresponding gauged SPT model, generalizing
the construction proposed in5. We note that the
relationship between these SPT phases and the string–
net models are explored via a different route also in6.
Finally we conclude in section VI and point to several
open problems.

拓扑绝缘体是一种新的量子物态。传统上固体材料可以按照其导电性质分为绝缘体和导体，其中绝缘体材料在其费米能处存在著有限大小的能隙，因而没有自由载流子；金属材料在费米能级处存在著有限的电子态密度，进而拥有自由载流子。而拓扑绝缘体是一类非常特殊的绝缘体，从理论上分析，这类材料的体内的能带结构是典型的绝缘体类型，在费米能处存在著能隙，然而在该类材料的表面则总是存在著穿越能隙的狄拉克型的电子态，因而导致其表面总是金属性的。拓扑绝缘体这一特殊的电子结构，是由其能带结构的特殊拓扑性质所决定的。
　　拓扑绝缘体研究现状： 
　　第一代， HgTe量子井 
　　第二代， BiSb 合金 
　　第三代， Bi2Se3， Sb2Te3， Bi2Se3 等化合物 
 
　　自旋轨道耦合引起了能带反转，以及材料表面的狄拉克型费米子 
　　根据理论预测，拓扑绝缘体于p波超导体界面将会形成majorana 费米子， 其特性符合量子计算机理论中的量子比特。


小薄膜大突破 人类或将进入拓扑量子计算时代 2014-01-07 11:27



（虽然人民日报内容一般都是很脑残，但这篇鉴定没事）

　　现在，人们看到的天气预报大致是这样的：某月某日，晴，6—15℃，西北风3-4级。未来，天气预报会变成这样：某月某日某分某秒，实时温度15℃，西北风3级。人们可以清楚地知道下一小时、下一分钟甚至是下一秒的天气实时情况。

　　有了拓扑量子计算机，像实时天气预报这样便捷的生活将会变成现实。

　　上海交通大学物理系贾金锋、钱冬研究组今天宣布，他们在实验室制备出一种由拓扑绝缘体材料和超导体材料复合而成的特殊人工薄膜。这种特殊的薄膜是产生Majorana费米子的必要条件。该团队有望在年内实现探测Majorana费米子的突破。

　　上海交大特别研究员钱冬介绍，如果找到了Majorana费米子，将使在固体中实现拓扑量子计算成为可能，这将引发未来电子技术的新一轮革命，人类也将进入拓扑量子计算时代。

　　有关“Majorana费米子”预言得验证 

　　粒子世界有两大人丁兴旺的“家族”：费米子和玻色子，是以物理学家费米和波色的名字命名的。费米子家族的典型代表是电子，它存在于我们日常使用的各种电器中；玻色子家族最常见的代表是光子，也就是我们熟悉的光。玻色子家族的共同脾性是：“翻脸”后还是一家人；费米子家族则完全相反，一旦“翻脸”就成“陌路人”。

　　物理学家认为，任何粒子都有它的双胞胎兄弟，也就是它的反粒子。1937年，意大利物理学家埃托雷·马约拉纳(Majorana)预测，自然界中可能存在一类特殊的费米子，它是自己的反粒子。人们将其命名为“Majorana费米子”。

　　“Majorana费米子”很神秘莫测。从20世纪到21世纪，全世界物理学家一直在寻找它。高能物理学家认为，中微子可能就是一种Majornana费米子。凝聚态（固体）物理学家们则在不同的材料体系中热情地寻找着Majorana费米子。理论物理学家提出了多个“Majorana费米子”可能“藏身”的材料体系，其中上海交大低维物理和界面工程实验室贾金锋、钱冬、刘灿华、高春雷四位教授联合攻关的拓扑绝缘体与超导体的界面，就是极有可能存在“Majorana费米子”的地方。

　　钱冬教授介绍，近年来，随着拓扑绝缘体的问世，国际上掀起了新一轮的在实验中追寻“Majorana费米子”的竞赛。上海交大已经制备出最适合探测和操纵“Majorana费米子”的人工薄膜系统，有望在年内实现探测新突破。届时，埃托雷·马约拉纳的跨世纪预言也将得到应验。

　　小薄膜有望成就物理学重大突破 

　　找到“Majorana费米子”，希望寄托于上海交大科研团队研究的一种特殊人工薄膜。这种神奇的薄膜，由拓扑绝缘体材料和超导材料复合而成。厚度只有一根头发丝的一万分之一。通过精确控制，将所需材料的原子一层一层的垒起来，最终达到产生“Majorana费米子”的必要要求。

　　钱冬形象化地把拓扑绝缘体比作是桌面，超导材料是桌子，怎么把台面和桌子有机地合在一块组合成一个更漂亮的桌子呢？这个看似简单的事情却是物理学领域的一个大难题。目前，国际上已经有多个研究组能够生长出高质量拓扑绝缘体薄膜，但由于界面反应和晶格匹配等问题，拓扑绝缘体与超导体之间的高质量的薄膜非常难以制备。

　　上海交大低维物理和界面工程实验室想出了一个解决方法，他们通过无数次实验，在拓扑绝缘体与超导体之间插入一种超薄的过渡层，从而形成了一种特殊的人工薄膜，首次成功地实现了超导体和拓扑绝缘体的“珠联璧合”。他们发现超导的特性能够传递到拓扑绝缘体上，使拓扑绝缘体也具有了超导体的“本领”。

　　小小薄膜成就了物理学领域的重大突破。这项研究成果即将在Science杂志发表，其网站已先行发布。该工作被Science审稿人评价为“材料科学的突破”和“巨大的实验成就”。

　　人类或将进入拓扑量子计算时代 

　　20世纪重大成就之一是计算机的发明。人类的工作、生活已离不开计算机。但我们现在使用的计算机还处于大规模集成电路时代。近半个世纪以来，计算机的性能价格比基本遵循着著名的摩尔定律：芯片的集成度和性能每18个月提高一倍。然而，随着半导体加工工艺进步，人们预期在不远的将来半导体集成电路中晶体管的尺寸将达到10纳米的尺度，而依靠提高集成电路的密集度来增加计算能力将不太可能。如何进一步提高计算能力，已是计算机发展面临的重大挑战。

　　用量子力学效应实现全新的计算模式，是一条正在探索的途径，这就是量子计算。量子计算机的运算空间比通常计算机大许多。但研究发现，存储在量子状态中的信息容易受到外界的影响而出错甚至丢失。传统计算机里面的一些容错方法也不适合于量子计算机。

　　因此，容错性成为实现量子计算的关键。而拓扑量子计算则是近年来发展的一个可能的解决方案。找到“Majorana费米子”，就仿佛找到了一把通往拓扑量子计算时代的钥匙，它使在固体中实现拓扑量子计算成为可能，人类也将进入拓扑计算时代。

　　钱冬介绍说，与通常的量子计算机不同，拓扑量子计算机中存储信息的量子状态受到额外的保护，这种量子态不受局域环境扰动。通常的量子计算机中，信息是存储在特定的位置，一个萝卜一个坑，坑坏了，萝卜也就没有了。而拓扑量子计算机中信息的存储是非局域的，“一个萝卜很多坑”，即使有几个坑坏掉，系统还能够根据其他的萝卜块情况得出整个萝卜的信息。拓扑量子计算是一种在硬件上容错的量子计算，它提供了通向固态量子计算的一条可行途径。



美研制奇特拓扑超导材料 表面金属内部超导体2014-01-07 11:29 (分类:默认分类)


资料图：量子计算机处理器(科学网-kexue.com配图)


3年前，美国普林斯顿大学的一个研究小组发现了三维拓扑绝缘体，这是一种金属表面的奇怪绝缘体，虽然它独特的属性具有很大应用潜力，但用于量子计算机却并非理想材料。两年来，科学家经过不断探索，完全扭转其性质，使之成为表面是金属、内部却具有超导性的拓扑超导体。这种新材料的发现有望发展出新一代电子学，使当前的信息存储与处理方式完全改观。
表面是金属内部是超导体
据美国物理学家组织网11月3日（北京时间）报道，普林斯顿大学扎西德·哈桑领导的研究小组发现了一种具有“双重性格”新型晶体材料：在极低温度下，晶体内部表现与普通超导体类似，能以零电阻导电；同时，它的表面是仍有电阻的金属，能传输电流。相关成果发表在最新一期《自然·物理学》杂志上。
实验中，为了评价新晶体材料的性能，研究人员利用X光谱进行分析，通过研究X射线轰击出来的单个电子来确定晶体的真实属性，测试发现生成的是一种拓扑超导体。研究人员进一步在晶体的表面发现了不同寻常的电子，其表现得像轻子。由于哈桑小组去年曾经第一次直接观察到了一种被称为螺旋状狄拉克费米子的电子，此时他们立刻认出了这种电子就是科学家长期寻找的马拉约那费米子（Majorana fermions）。
而宾夕法尼亚大学物理学家查尔斯·凯恩预测，如果一种拓扑超导体取代了一种拓扑绝缘体，把这种混合材料置于强磁场中时，其边界电子将变成马拉约那费米子。由于这种新晶体材料囊括了金属、绝缘体和传统超导体等多重“身份”，如何根据电子状态来将它归类让科学家困惑不已。哈桑表示，拓扑超导体除了表面是金属以外，其他部分都是超导体，这将给我们带来许多应用前景。
把绝缘体变成超导体
2007年，哈桑领导的研究小组发现了三维拓扑绝缘体硒化铋。在过去的两年中，研究小组扭转了硒化铋的属性，使其变成了表面是金属、内部为超导体的材料，这种属性就很适合于未来电子学的开发。
为了使超导体具有拓扑性质，参与研究的普林斯顿大学化学教授罗伯特·卡瓦把铜原子嵌入硒化铋半导体的原子晶格中，发明了一种新晶体。这一过程称为半导体掺杂，是一种改变材料电子数量的方法，用来转变其电性。结果发现，在低于4K（约零下269摄氏度）的温度下，合适的嵌入数量能将晶体转变成一种超导体。但美中不足的是，根据最初的实验结果，超导体无法长久保持其拓扑性质，在真空中仅能保持几个月。
加州大学伯克利分校物理副教授约尔·摩尔说，从理论上而言，如果一种拓扑绝缘体变成了拓扑超导体，它会具有一些超常的性质，最异类的就是出现马拉约那费米子。由普通原子核和电子构成的固体能“生成”具有特异性质的粒子，比如分数电荷，但马拉约那费米子是零质量零电荷，这可能是最奇怪的。尽管还没有能检测拓扑超导体的工具，但哈桑的研究在正确的方向上迈进了一大步。
应用还需再等几十年
量子计算机使用次原子粒子“量子”来存储和处理信息。量子计算机将来能以远远超过今天传统计算机的速度来操作数据，然而，研制更高性能量子计算机的努力，却由于量子行为的不确定而受到阻碍。如果多个马拉约那费米子的运动能被预测，拓扑量子计算机用它们来存储信息将是容错的，即计算机能“知道”自己在执行对错计算时是否出现了错误。
“从新物理学发展到新技术应用需要很长时间，通常要20年到30年时间。”哈桑介绍说，拓扑超导体最激动人心的应用就是高能量子计算机，它能在计算中发现错误，一旦出错就会在信息处理过程中产生抵抗。他解释说，普通电子带负电荷，而马拉约那费米子是中性的，它不会被附近的粒子、原子吸引或排斥，它们的行动就是可预测的，有着预定的轨迹，这是它们真正的潜能所在。
哈桑也称，这种具有双重电子特性的新型超导材料，可以被认为是一种特殊的绝缘体。“我们可以利用这一点来‘哄骗’电子嗖嗖地跑到它的表面上，变成马拉约那费米子。”
“这些超导体是产生和控制马拉约那费米子的理想育儿所。”论文第一作者L·安德鲁·雷说。由于粒子是存在于超导体中的，能以低能耗装置来控制，不仅环保，也避免了当前硅材料不可避免的过热问题。
目前，研究小组还在鉴别其他种类的拓扑超导体和拓扑绝缘体。关于进一步的研究，哈桑和他的团队表示将继续检测马拉约那费米子，找出控制它们性质的方法。他们的两个重要目标，一是找到高温超导的拓扑材料，二是开发内部高度绝缘的拓扑绝缘体。
量子计算机之所以成为人们关于这种新材料应用的第一联想，主要是因为如果用马拉约那费米子承载量子信息，将能有效保护脆弱的量子态不受侵害。与两次受到诺贝尔奖青睐的量子霍尔效应极其相似，“非驴非马”的拓扑超导体和不久前发现的拓扑绝缘体展示的奇异量子现象自然让这些科学家兴奋不已。拓扑绝缘体一经发现便成为宠儿，拓扑超导体肯定更将被物理学界视作明天娇子。求索之路遥遥，即使它只是昙花一现，我们无疑也会从中获知良多。


超冷原子模拟“量子二极管”




图片标题：单向费米子系统
图注：图为麻省理工大学的研究人员制作的自旋二极管（spin diode）的艺术效果图。自旋方向为顺时针的原子只沿一个方向运动，而自旋方向为逆时针的原子则朝反方向运动。
两个彼此独立的物理研究小组首次利用超冷费米子原子模拟了“自旋轨道的耦合”，这一相互作用在固体材料的电子性质中占有重要地位。这两组实验都是将激光束打向原子，从而改变其动量，动量改变的多少则由原子的固有自旋决定。由于该项模拟中原子间的相互作用可以被精确地操控，因此这项突破性的进展将为一系列物理研究带来新的进展，其中包括磁学、拓扑绝缘体以及马约拉纳费米子（Majorana fermions，编注：这类费米子的反粒子即它本身，而狄拉克费米子则不同于其自身的反粒子）。
自旋轨道耦合描述的是材料中电子的固有自旋以及离子的环绕运动所引起的磁场，这两者之间的相互作用。自旋轨道耦合对材料的磁场性质起着重要作用，因此它也影响着“自旋电子学”仪器的性能。这类仪器所利用的并非电子的电荷量而是电子的自旋值，而且将来它们极有可能为我们带来更快更高效的计算机。
量子模拟器
基于超冷原子云的物理属性，物理学家们常常利用它来模拟自旋轨道的耦合现象。这些“量子模拟器”是通过将原子气体置于激光和磁场之下予以实现的，研究人员利用它可以模拟出固体中电子所受到的相互作用力。这类模拟器的优势在于，和在固体中不同，这些相互作用的强度可以很方便地进行调节，使物理学家们得以检验凝聚态物理的若干理论。
2011年美国马里兰州国家标准技术研究所（NIST）的兰·斯贝尔曼（lan Spielman）首次利用极冷玻色原子气模拟了自旋轨道耦合。现在，两个相互独立的研究小组将斯贝尔曼的技术扩展运用到了费米子上，这两个小组一个是由中国清华大学的翟荟和山西大学的张靖领导的，另一个是由美国麻省理工大学的马丁·茨莱维因（Martin Zwierlein）和劳伦斯·卓（LawrenceCheuk）领导的。因为电子都是费米子而非玻色子，所以这项新的工作更多是与电子物理学相关的。
钾-40原子
中国的研究小组首先将两百万个钾-40原子囚禁在一个光学势阱中，并冷却到系综的费米温度以下。这意味着原子气中几乎所有的原子都处于可能的最低能级上，就如同金属中的传导电子一样。研究小组所关注的是两个挨得很近的磁能能级，并用这两个能级来模拟电子的自旋，一个能级相当于自旋向上，另一个则相当于自旋向下。
研究人员随后从两个相反的方向朝原子气体发射两束激光，并将激光设置得与两个自旋态间的跃迁形成共振，所谓跃迁就是指原子连续不断地吸收和放出光子的过程。由于这些光子携带有动量，所以如果一个原子吸收的光子方向同再次放出光子的方向相同，那么原子的动量将保持不变。然而原子也可以被来自相反方向的光束所激发，并辐射光子，从而改变原子的动量。这一相互作用包含了原子自旋方向的改变，因此也可以类推为自旋轨道的耦合，虽然只是一维情况下的。
中国的研究小组采用这一系统研究了自旋轨道耦合的一些性质。在一个实验中，研究人员首先选择自旋方向都指向同一个方向的起始态，然后在很短的时间内（大约几百个微秒）向其发射激光脉冲以开启自旋轨道间的耦合作用。他们发现在经过一个称为“移相（dephasing）”的过程后，自旋开始指向了不同的方向。这是费米子的预期结果，因为具有相同自旋值的原子不可能具有相同的动量，所受到的自旋轨道相互作用也就不同。
理解移相过程十分重要，因为它阻碍了将自旋性质应用到自旋电子学以及量子计算机等技术实用当中。研究小组还研究了自旋轨道耦合其他方面的作用效果，包括它对原子动量分布的影响。
锂-6原子
与此同时，麻省理工大学的物理学家采用的是锂-6原子气，这意味着他们要实现自旋轨道耦合要比中国的研究团队更困难一些。其困难在于像锂原子这样的轻原子在对光的共振吸收过程中更容易被加热。为了解决这一难题，麻省理工的研究小组将大多数的原子都保持在“蓄积能级（reservoir states）”中，这些能级上的原子不与入射光发生相互作用，因而也能保持较低的温度，然后再采用电波将一小部分原子引入到进行自旋轨道耦合的能级上。
麻省理工研究小组的关注点是超冷原子可以用于模拟“自旋二极管（spin diode）”，所谓自旋二极管就是使自旋向上的原子只进不退，而自旋向下的原子只退不进，这种装置很可能在自旋电子回路的发展中扮演十分关键的角色。卓教授说道：“原子气扮演的正是量子二极管，它可以操控自旋电流的流向。”
模拟带状结构
通过将射频辐射技术运用到原子气当中，麻省理工的物理学家模拟出了同一维晶格相同的周期性势场。在实际材料中，周期性的势场将导致自旋能带（spin-dependent energy band）的出现。据研究小组所说，能够用这种方法产生出自旋能带结构，意味着可以模拟出拓扑绝缘体。
两个研究团队所模拟的自旋轨道耦合都只是一维上的，因而这些方法还不能模拟大多数电子器件中二维或者三维的系统。不过在一维的系统中也有一些有趣的现象值得研究。例如，可以模拟半导体/超导纳米线（nanowire）中的电子行为。这类系统被认为可以暂存像马约拉纳费米子这样的准粒子，马约拉纳费米子是它自己的反粒子，一直备受关注。
这两组实验都刊登在《物理评论快报（Physical Review Letters）》上。
关于作者:
哈米斯·约翰斯顿（Hamish Johnston）是物理世界网站（physicsworld.com）的一名编辑。


《物理评论B》：拓扑超导体与拓扑半金属研究获进展2014-01-07 11:24 (分类:默认分类)


导读：图1，NaCoO2的表面模型图2，5-层 NaCoO2的自旋极化能带图，自旋向下为绝缘态，只有自旋向上的费米面，其自旋构型与拓扑绝缘体表面态相似。图3，在HgCr2Se4侧表面上形成的“费米弧”图4， HgCr2Se4中霍尔电导随量子阱厚度d的台阶式跳变近年来，作为一种新的量子物态，拓扑绝缘体因其丰富奇特的电子特性以及在未来电子技术中……


图1，NaCoO2的表面模型

图2，5-层 NaCoO2的自旋极化能带图，自旋向下为绝缘态，只有自旋向上的费米面，其自旋构型与拓扑绝缘体表面态相似。

图3，在HgCr2Se4侧表面上形成的“费米弧”

图4， HgCr2Se4中霍尔电导随量子阱厚度d的台阶式跳变
近年来，作为一种新的量子物态，拓扑绝缘体因其丰富奇特的电子特性以及在未来电子技术中的应用前景，在世界范围内取得了快速发展，并成为凝聚态物理研究中的一个热点领域。在寻找具有更高应用价值的强拓扑绝缘体材料的同时，许多新的拓扑物性被预言和发现，如磁单极、拓扑超导态、Majorana费米子和量子化的反常霍尔效应等。其中，尤以Majorana费米子和量子反常霍尔效应因其奇特的量子特性和应用价值而备受关注。
1937年，意大利物理学家Ettore Majorana提出一组满足Dirac方程的实数解，与描述电子的复数解不同，这组实数解描述了一个具有电中性，且该粒子的反粒子就是其自身的全新粒子，后来被人们称为Majorana费米子。理论研究发现，Majorana费米子满足非阿贝尔统计规律，即操作粒子的结果与操作的先后顺序有关，这为设计新概念的拓扑量子计算机提供了重要途径。因而，寻找Majorana费米子既有重要的基础理论意义，也有巨大的潜在应用价值。
中国科学院物理研究所/北京凝聚态物理国家实验室的方忠研究组及其合作者，在该研究方向上取得重要进展，从理论上预言NaCoO2的表面态具有半金属性，在其与传统s波超导体的界面上可以诱导出拓扑超导态。NaxCoO2是层状结构材料〔图1〕，当x大于0.5而小于1.0时，CoO2层具有铁磁序而相邻层反铁磁排列；x=1.0时，NaCoO2是简单的能带绝缘体。翁红明、徐刚等人巧妙地利用这一特点，提出如果保持体内Na的浓度x=1.0，而部分或全部除去最外层的Na〔图1〕，通过理论计算发现，表面的CoO2层具有稳定的铁磁半金属性，而且只有单个自旋极化的费米面〔图2〕。由于Rashba型自旋轨道耦合效应，其费米面具有与拓扑绝缘体表面态费米面类似的自旋构型〔图2〕，因而可以用来实现p+ip型的拓扑超导态，并进而可能在其中找到Majorana费米子态。本工作以Rapid Communicaton形式发表在Phys. Rev. B 84, 060408 (2011)，并被选为编辑推荐的文章。
最近，徐刚、翁红明等人又将凝聚态中电子态通过拓扑分类的概念从绝缘体推广到了半金属。他们通过计算发现了一种特殊的拓扑半金属态材料——HgCr2Se4。HgCr2Se4具有典型的尖晶石结构，它的低能电子结构可以很好地用我们熟悉的重空穴、轻空穴和具有S轨道特性的导带来描写。在低温下，Cr离子的磁矩形成很强的铁磁态，费米面附近的能带感受到很强的塞曼劈裂，这导致了自旋向下能带反转而自旋向上的能带维持正常的结构。所以在HgCr2Se4材料中，只有自旋取向跟磁化方向一致的那一半能带形成了反带结构，从而导致所谓的既是单自旋金属（暂译，half－metal）又是半金属（semi－metal）的极为特殊的电子结构。在这种特殊的电子结构下，体系的能带在沿Z轴的两个互为反演的点上交叉，形成所谓的“Weyl”费米子的特殊结构，“Weyl”费米子是狄拉克费米子的一半，在空间维度是三维的情况下，任何保持平移对称的微扰项都不能使得能隙打开，而只能使交叉点在k空间内移动。因此，这样的“Weyl”费米子体系是拓扑稳定的。
徐刚等人在文章中进一步对该体系的拓扑结构进行了分析，指出这类“Weyl”费米子体系可以通过研究有效Chern数随着z方向动量演化来很好地刻画。“Weyl”费米子的一个重要的物理后果是在其侧表面上形成所谓的“费米弧”（见图 3），即不连续的费米面结构。这完全是其特殊的能带拓扑结构所导致的。同时，徐刚等人进一步预言，如果把HgCr2Se4材料制成有限厚度的量子阱，可以实现量子化的反常Hall效应（图 4）。该文章在Phys. Rev. Lett.发表以后，引起了编辑部的浓厚兴趣，他们专门约稿在Physics杂志上同步刊登了介绍文章，向读者重点推荐该工作，同时该文也被PRL编辑部评为当期的编辑推荐文章。
以上工作得到了中国科学院、国家自然科学基金委、科技部国家重点基础研究发展计划、重大科学研究计划和国际科技合作计划的支持。（来源：中科院物理研究所）




生活大爆炸里也有出线topological insulator，分享一篇文章


TBBT第4季14集: 戏剧催化课（附视频）摇滚驴 2011-04-03 12:25
（文/大卫·萨尔兹保）石墨烯已经过时了，现在是碲化物的时代。
在谢耳朵的讲座里，他和学生们（还有1500万观众）分享了最近在铋、碲和锡的某些化合物中发现的奇怪现象。奇怪的新物质既是导体也是绝缘体。这些碲化物兄弟们都是最近发现的新材料家族的一份子，正如谢耳朵所说，这个家族名叫拓扑绝缘体。
在典型塑料材料中，基本结构的电子固定不动。因为这些材料可以防止导体短路，所以被称作绝缘体。相对于最好的导电体，绝缘材料电子的导电性为10的26次方分之一，也就是10,000,0000,000,000,000,000,000,000分之一。物理学中很少能看到如此庞大的数量级变化。
今晚的白板上，观众朋友们看到了碲化铋、碲化镉、碲化汞客串出场。这些材料总体绝缘但表面却导电。这是为什么呢？
聪明的年轻人可能已经想到通过电镀某些塑料可以得到同样的效果。这是理查德•费曼的研究成果之一。但是这和新材料的原理截然不同，因为它要用两种材料才能达到同样的效果。物理学家们从未想过一种材料就可以产生这个现象。
新材料与常规绝缘体的不同之处在于“拓扑”。这个表述来源于数学的分支——拓扑学，一项研究物体基本形状的学科。你可以把一个甜甜圈伸展成咖啡杯（一个孔），但无法把它变成带有两个孔的物体。同样的，常规绝缘体电子轨道的基本结构可由一个简单的环表示，而“拓扑”环与最简单结构完全不同——它是三叶结结构。

拓扑绝缘体的电子轨道结构类似于三叶结
电子自旋和轨道角动量的相互作用形成了三叶结的数学结构。而你们的科学顾问也无法理解为什么这种材料表面会发生如此明显的变化：拓扑绝缘体变得与金属类似并且导电，而常规绝缘体却仍不导电。如果你们有好的想法，那么就请留言吧！
科学家们曾观测到相似的效应是在难以建立的平面结构中。不过，就像好莱坞一样，物理学家们因为碲化铋的到来步入了三维时代。拓扑绝缘体可以通过标准的半导体制造工艺生成。既导电又绝缘的拓扑绝缘体不是“温室”的花朵，只能在真空或强冷却的昂贵实验室中生存。这些材料的性质在室温下也维持不变，哪怕是放在实验室的板凳上或是在手中把玩。
近五年类似的研究持续升温，科学家们发现了许多其他化合物，它们不仅带有拓扑绝缘体的双重属性，而且通过铋、硒、铜的晶体结构还可以实现超导，电子在移动中没有任何损耗。
拓扑绝缘体使很多我们从未想象到的新型计算和材料的应用成为现实。它们的外在表现和本质都令物理学家着迷。可悲的是一些主流文章掉入了磁单极子的谬误中。谢耳朵在全国公共广播电台科学星期五栏目中向主持人伊拉•弗莱托做了同样的吐槽。最新的文章表示拓扑绝缘体吸引人之处在于可以生成轴子的准粒子模型，与粒子物理学中找寻的物质类似。然而，同磁单极子论一样，这篇文章也遗漏了重要的一点：粒子物理学家发现新粒子不只是为了观察它们的数学行为方式。我们找寻是因为它们诠释了宇宙。同轴子一样，它可以验证某些自然中存在深对称性的理论。轴子可能是星系中的暗物质，但是在凝聚态系统中发现的类轴子物质不是真正的轴子。它与轴子完全不同。当然拓扑绝缘体仍令人振奋。正因为作者们对新材料前景的错误论述，这些主流文章不仅深深地误导了读者而且还对新材料的发展造成了负面影响。
不过，拓扑绝缘体的前途一片光明，我们期待着下次谢耳朵的白板会出现在斯德哥尔摩的讲堂中。（如果大家像谢耳朵的学生一样，觉得这篇博文十分无聊……就来点击下面的回应按钮吐槽吧。）
原文看这里
科技名博微博
博主介绍： 大卫·萨尔兹保（David Saltzberg）是美国加州大学洛杉矶分校的物理、天文学教授。与此同时，他还担当着《生活大爆炸》的科学顾问。剧中Sheldon一伙所说的那些专业术语，全部出自此人之手。在他的博客里又进一步阐释了那些令人挠头的科学小知识。