Quantum Field Theory II

Fourier analysis[edit]

One of the basic goals of Fourier analysis is to decompose a function into a (possibly infinite) linear combination of given basis functions: the associated Fourier series. The classical Fourier series associated to a function f defined on the interval [0, 1] is a series of the form

where

The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The basis functions are sine waves with wavelengths λ/n (n=integer) shorter than the wavelength λ of the sawtooth itself (except for n=1, the fundamental wave). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation of the summed terms about the sawtooth is called the Gibbs phenomenon.

A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f. Hilbert space methods provide one possible answer to this question.^[33] The functions e_n(θ) = e^2πinθ form an orthogonal basis of the Hilbert spaceL²([0,1]). Consequently, any square-integrable function can be expressed as a series

and, moreover, this series converges in the Hilbert space sense (that is, in the L² mean).

The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space.^[34] The abstraction is especially useful when it is more natural to use different basis functions for a space such as L²([0,1]). In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials or wavelets for instance,^[35] and in higher dimensions into spherical harmonics.^[36]

For instance, if e_n are any orthonormal basis functions of L²[0,1], then a given function in L²[0,1] can be approximated as a finite linear combination^[37]

The coefficients {a_j} are selected to make the magnitude of the difference ||ƒ − ƒ_n||² as small as possible. Geometrically, the best approximation is the orthogonal projection of ƒonto the subspace consisting of all linear combinations of the {e_j}, and can be calculated by^[38]

That this formula minimizes the difference ||ƒ − ƒ_n||² is a consequence of Bessel's inequality and Parseval's formula.

In various applications to physical problems, a function can be decomposed into physically meaningful eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral study of functions, in reference to the spectrum of the differential operator.^[39] A concrete physical application involves the problem of hearing the shape of a drum: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself?^[40] The mathematical formulation of this question involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.

Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis, as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.

Quantum mechanics[edit]

In the mathematically rigorous formulation of quantum mechanics, developed by John von Neumann,^[41] the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called state vectors) residing in a complex separable Hilbert space, known as the state space, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each observable is represented by a self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.

The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution.

The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.

For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.

Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute, and gives a specific form that thecommutator must have.

Properties[edit]

Pythagorean identity[edit]

Two vectors u and v in a Hilbert space H are orthogonal when  = 0. The notation for this is u ⊥ v. More generally, when S is a subset in H, the notation u ⊥ S means that u is orthogonal to every element from S.
When u and v are orthogonal, one has

By induction on n, this is extended to any family u₁,...,u_n of n orthogonal vectors,

Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series. A series Σ u_k oforthogonal vectors converges in H  if and only if the series of squares of norms converges, and

Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.

Parallelogram identity and polarization[edit]

By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds:

Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity.^[42] For real Hilbert spaces, the polarization identity is

For complex Hilbert spaces, it is

The parallelogram law implies that any Hilbert space is a uniformly convex Banach space.^[43]

Best approximation[edit]

If C is a non-empty closed convex subset of a Hilbert space H and x a point in H, there exists a unique point y ∈ C that minimizes the distance between x and points in C,^[44]

This is equivalent to saying that there is a point with minimal norm in the translated convex set D = C − x. The proof consists in showing that every minimizing sequence (d_n) ⊂ D is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point in D that has minimal norm. More generally, this holds in any uniformly convex Banach space.^[45]

When this result is applied to a closed subspace F of H, it can be shown that the point y ∈ F closest to x is characterized by^[46]

This point y is the orthogonal projection of x onto F, and the mapping P_F : x → y is linear (see Orthogonal complements and projections). This result is especially significant inapplied mathematics, especially numerical analysis, where it forms the basis of least squares methods^{[citation needed]}.

In particular, when F is not equal to H, one can find a non-zero vector v orthogonal to F (select x not in F and v = x − y). A very useful criterion is obtained by applying this observation to the closed subspace F generated by a subset S of H.
A subset S of H spans a dense vector subspace if (and only if) the vector 0 is the sole vector v ∈ H orthogonal to S.

Duality[edit]

The dual space H* is the space of all continuous linear functions from the space H into the base field. It carries a natural norm, defined by

This norm satisfies the parallelogram law, and so the dual space is also an inner product space. The dual space is also complete, and so it is a Hilbert space in its own right.

The Riesz representation theorem affords a convenient description of the dual. To every element u of H, there is a unique element φ_u of H*, defined by

The mapping  is an antilinear mapping from H to H*. The Riesz representation theorem states that this mapping is an antilinear isomorphism.^[47] Thus to every element φof the dual H* there exists one and only one u_φ in H such that

for all x ∈ H. The inner product on the dual space H* satisfies

The reversal of order on the right-hand side restores linearity in φ from the antilinearity of u_φ. In the real case, the antilinear isomorphism from H to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.

The representing vector u_φ is obtained in the following way. When φ ≠ 0, the kernel F = Ker(φ) is a closed vector subspace of H, not equal to H, hence there exists a non-zero vector v orthogonal to F. The vector u is a suitable scalar multiple λv of v. The requirement that φ(v) = ⟨v, u⟩ yields

This correspondence φ ↔ u is exploited by the bra–ket notation popular in physics. It is common in physics to assume that the inner product, denoted by ⟨x|y⟩, is linear on the right,

The result ⟨x|y⟩ can be seen as the action of the linear functional ⟨x| (the bra) on the vector  |y⟩ (the ket).

The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the topological dual of any inner product space can be identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert space His reflexive, meaning that the natural map from H into its double dual space is an isomorphism.

Weakly convergent sequences[edit]

Main article: Weak convergence (Hilbert space)

In a Hilbert space H, a sequence {x_n} is weakly convergent to a vector x ∈ H when

for every v ∈ H.

For example, any orthonormal sequence {f_n} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {x_n} is bounded, by the uniform boundedness principle.

Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem).^[48] This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on R^d. Among several variants, one simple statement is as follows:^[49]
If f: H → R is a convex continuous function such that f(x) tends to +∞ when ||x|| tends to ∞, then f admits a minimum at some point x0 ∈ H.
This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space H are weakly compact, since H is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.

Banach space properties[edit]

Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces.^[50] The open mapping theorem is equivalent to the closed graph theorem, which asserts that a function from one Banach space to another is continuous if and only if its graph is a closed set.^[51] In the case of Hilbert spaces, this is basic in the study of unbounded operators (see closed operator).

The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if y is the element of a closed convex set F closest to x, then the separating hyperplane is the plane perpendicular to the segment xy passing through its midpoint.^[52]

Operators on Hilbert spaces[edit]

Bounded operators[edit]

The continuous linear operators A : H₁ → H₂ from a Hilbert space H₁ to a second Hilbert space H₂ are bounded in the sense that they map bounded sets to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operator norm given by

The sum and the composite of two bounded linear operators is again bounded and linear. For y in H₂, the map that sends x ∈ H₁ to ⟨Ax, y⟩ is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form

for some vector A* y in H₁. This defines another bounded linear operator A*: H₂ → H₁, the adjoint of A. One can see that A** = A.

The set B(H) of all bounded linear operators on H, together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type ofoperator algebra.

An element A  of B(H) is called self-adjoint or Hermitian if A*= A. If A  is Hermitian and ⟨Ax, x⟩ ≥ 0 for every x, then A  is called non-negative, written A ≥ 0; if equality holds only when x = 0, then A  is called positive. The set of self adjoint operators admits a partial order, in which A ≥ B if A − B ≥ 0. If A  has the form B* B  for some B, then A is non-negative; if B is invertible, then A  is positive. A converse is also true in the sense that, for a non-negative operator A, there exists a unique non-negative square root B such that

In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element A of B(H) is called normal if A* A = A A*. Normal operators decompose into the sum of a self-adjoint operators and an imaginary multiple of a self adjoint operator

that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.

An element U  of B(H) is called unitary if U  is invertible and its inverse is given by U*. This can also be expressed by requiring that U  be onto and ⟨Ux, Uy⟩ = ⟨x, y⟩ for all x andy in H. The unitary operators form a group under composition, which is the isometry group of H.

An element of B(H) is compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator T is compact if, for any bounded sequence {x_k}, the sequence {Tx_k} has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations. Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel. The index of a Fredholm operator T is defined by

The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem.

Unbounded operators[edit]

Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics.^[53] An unbounded operator T on a Hilbert space H is defined as a linear operator whose domain D(T) is a linear subspace of H. Often the domain D(T) is a dense subspace of H, in which case T is known as a densely defined operator.

The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of theobservables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space L²(R) are:^[54]

• A suitable extension of the differential operator
  
  

where i is the imaginary unit and f is a differentiable function of compact support.

• The multiplication-by-x operator:
  
  

These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of L²(R).

Constructions[edit]

Direct sums[edit]

Two Hilbert spaces H₁ and H₂ can be combined into another Hilbert space, called the (orthogonal) direct sum,^[55] and denoted

consisting of the set of all ordered pairs (x₁, x₂) where x_i ∈ H_i, i = 1,2, and inner product defined by

More generally, if H_i is a family of Hilbert spaces indexed by i ∈ I, then the direct sum of the H_i, denoted

consists of the set of all indexed families

in the Cartesian product of the H_i such that

The inner product is defined by

Each of the H_i is included as a closed subspace in the direct sum of all of the H_i. Moreover, the H_i are pairwise orthogonal. Conversely, if there is a system of closed subspaces,V_i, i ∈ I, in a Hilbert space H, that are pairwise orthogonal and whose union is dense in H, then H is canonically isomorphic to the direct sum of V_i. In this case, H is called the internal direct sum of the V_i. A direct sum (internal or external) is also equipped with a family of orthogonal projections E_i onto the ith direct summand H_i. These projections are bounded, self-adjoint, idempotent operators that satisfy the orthogonality condition

The spectral theorem for compact self-adjoint operators on a Hilbert space H states that H splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.

Tensor products[edit]

Main article: Tensor product of Hilbert spaces

If H₁ and H₂, then one defines an inner product on the (ordinary) tensor product as follows. On simple tensors, let

This formula then extends by sesquilinearity to an inner product on H₁ ⊗ H₂. The Hilbertian tensor product of H₁ and H₂, sometimes denoted by , is the Hilbert space obtained by completing H₁ ⊗ H₂ for the metric associated to this inner product.^[56]

An example is provided by the Hilbert space L²([0, 1]). The Hilbertian tensor product of two copies of L²([0, 1]) is isometrically and linearly isomorphic to the space L²([0, 1]²) of square-integrable functions on the square [0, 1]². This isomorphism sends a simple tensor  to the function

on the square.

This example is typical in the following sense.^[57] Associated to every simple tensor product x₁ ⊗ x₂ is the rank one operator from H₁^∗ to H₂ that maps a given  as

This mapping defined on simple tensors extends to a linear identification between H₁ ⊗ H₂ and the space of finite rank operators from H*₁ to H₂. This extends to a linear isometry of the Hilbertian tensor product  with the Hilbert space HS(H*₁, H₂) of Hilbert–Schmidt operators from H*₁ to H₂.

Orthonormal bases[edit]

The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces.^[58] In a Hilbert space H, an orthonormal basis is a family {e_k}_k ∈ B of elements of H satisfying the conditions:

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[*]Orthogonality: Every two different elements of B are orthogonal: ⟨e_k, e_j⟩= 0 for all k, j in B with k ≠ j.

[*]Normalization: Every element of the family has norm 1:||e_k|| = 1 for all k in B.

[*]Completeness: The linear span of the family e_k, k ∈ B, is dense in H.

[/list]

A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if B is countable). Such a system is always linearly independent. Completeness of an orthonormal system of vectors of a Hilbert space can be equivalently restated as:
if ⟨v, ek⟩ = 0 for all k ∈ B and some v ∈ H then v = 0.
This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if S is any orthonormal set and v is orthogonal to S, then v is orthogonal to the closure of the linear span of S, which is the whole space.

Examples of orthonormal bases include:

• the set {(1,0,0), (0,1,0), (0,0,1)} forms an orthonormal basis of R³ with the dot product;
  
  
• the sequence {f_n : n ∈ Z} with f_n(x) = exp(2πinx) forms an orthonormal basis of the complex space L²([0,1]);
  
  

In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.

Sequence spaces[edit]

The space ℓ ² of square-summable sequences of complex numbers is the set of infinite sequences

of complex numbers such that

This space has an orthonormal basis:

More generally, if B is any set, then one can form a Hilbert space of sequences with index set B, defined by

The summation over B is here defined by

the supremum being taken over all finite subsets of B. It follows that, for this sum to be finite, every element of ℓ ²(B) has only countably many nonzero terms. This space becomes a Hilbert space with the inner product

for all x and y in ℓ ²(B). Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.

An orthonormal basis of ℓ ²(B) is indexed by the set B, given by

Bessel's inequality and Parseval's formula[edit]

Let f₁, …, f_n be a finite orthonormal system in H. For an arbitrary vector x in H, let

Then ⟨x, f_k⟩ = ⟨y, f_k⟩ for every k = 1, …, n. It follows that x − y is orthogonal to each f_k, hence x − y is orthogonal to y. Using the Pythagorean identity twice, it follows that

Let {f_i }, i ∈ I, be an arbitrary orthonormal system in H. Applying the preceding inequality to every finite subset J of I gives the Bessel inequality^[59]

(according to the definition of the sum of an arbitrary family of non-negative real numbers).

Geometrically, Bessel's inequality implies that the orthogonal projection of x onto the linear subspace spanned by the f_i has norm that does not exceed that of x. In two dimensions, this is the assertion that the length of the leg of a right *** may not exceed the length of the hypotenuse.

Bessel's inequality is a stepping stone to the more powerful Parseval identity, which governs the case when Bessel's inequality is actually an equality. If {e_k}_k ∈ B is an orthonormal basis of H, then every element x of H may be written as

Even if B is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of countably many nonzero terms. This sum is called the Fourier expansion of x, and the individual coefficients ⟨x,e_k⟩ are the Fourier coefficients of x. Parseval's formula is then

Conversely, if {e_k} is an orthonormal set such that Parseval's identity holds for every x, then {e_k} is an orthonormal basis.

Hilbert dimension[edit]

As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space.^[60] For instance, since ℓ²(B) has an orthonormal basis indexed by B, its Hilbert dimension is the cardinality of B (which may be a finite integer, or a countable or uncountable cardinal number).

As a consequence of Parseval's identity, if {e_k}_k ∈ B is an orthonormal basis of H, then the map Φ : H →  ℓ²(B) defined by Φ(x) = (⟨x,e_k⟩)_k∈B is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that

for all x and y in H. The cardinal number of B is the Hilbert dimension of H. Thus every Hilbert space is isometrically isomorphic to a sequence space  ℓ²(B) for some set B.

Historical remarks. In 1954, the young physicists Yang and Mills tried to
generalize the Maxwell equations. They used an idea of Hermann Weyl published
in 1929.
• Weyl formulated the Maxwell equations as a gauge theory based on the commutative
Lie group U(1).
• The goal of Yang and Mills was to replace the commutative group U(1) by the
noncommutative Lie group SU(2).
To this end, they replaced the relation
Fjk = ∂jAk − ∂kAj, j= 1, 2, 3, 4
between the electromagnetic field tensor {Fij} and the 4-potential {Aj} by a modified
relation of the type
Fjk = ∂jAk − ∂kAj + AjAk −AkAj, j,k= 1, 2, 3, 4, (5.98)
where the complex 2 × 2 matrices A1,A2,A3,A4 are elements of the Lie algebra
su(2) of the symmetry group SU(2). Dianzhou Zhang asked Professor Yang in
an interview: An interesting question is whether you understood in 1954 the
tremendous importance of your joint paper with Mills on noncommutative gauge
theory. Yang answered:
No. In the 1950s we felt our work was elegant. I realized its importance in
the 1960s and its great importance to physics in the 1970s. Its relationship
to deep mathematics became only clear to me after 1974.
In the 1960s and the early 1970s, equations of the type (5.98) above were used in
order to formulate the Standard Model in elementary particle physics. Here, the
symmetry group U(1) × SU(2) × SU(3) is used.
• The crucial field tensor {Fij} describes all the 12 particles (i.e., the photon, 8
gluons, and 3 vector bosons) which are responsible for the interaction
• between the 12 basic particles (i.e., 6 quarks and 6 leptons – the electron, 3
neutrinos, the muon, and the tau) and their antiparticles.
The typical difficulty of any gauge theory is the fact that the interacting particles
are massless at the very beginning. In contrast to this, the three vector bosons
observed in nature are quite heavy – their masses equal about 100 proton masses.
One needs an additional field – the Higgs field – in order to generate the masses
of the vector bosons. This so-called Higgs mechanism will be thoroughly studied in
Vol. III.
In the late 1960s, Yang discovered the relation between Yang–Mills theory and
Riemannian geometry. He reports in the same interview as quoted above:
In the late 1960s, I began a new formulation of gauge fields, through the
approach of non-integrable phase factors. It happend that one semester, I
was teaching general relativity, and I noticed that the formula (5.93) above
in gauge theory and the formula (5.94) above in Riemannian geometry
are not just similar – they are, in fact the same if one makes the right
identification of symbols.
Yang continues:
With an appreciation of the geometrical meaning of gauge theory, I consulted
Jim Simons, a distinguished geometer, who was then the chairman
of the Mathematics Department at Stony Brook (Long Island, New York).
He said gauge theory must be related to connections on fiber bundles. I
then tried to understand fiber bundle theory from such books as Steenrod’s
The Topology of Fiber Bundles, Princeton University Press, 1951,
but I learned nothing. The language of modern mathematics is too cold
and abstract for a physicist.61
In 1975, Wu and Yang wrote a paper about global gauge theory. In this paper, they
published a quite interesting dictionary about the completely different terminology
of mathematicians and physicists concerning the same topic. For example:
connection in mathematics ⇔ potential in physics,
curvature in mathematics ⇔ field tensor (interaction) in physics,
structural equation ⇔ field tensor − potential relation,
change of bundle coordinates ⇔ gauge transformation,
structure group ⇔ gauge group.
In a long historical process, mathematicians tried to understand curvature, but
physicists studied the forces acting in the universe. Nowadays we know that mathematicians
and physicists did the same from an abstract mathematical point of view.
Observe that
The Cartan curvature 2-form is more fundamental than the Riemann curvature
tensor.
In fact, since RP (a, b, u, v) := −<F(a, b)u|v>P for all a, b, u, v ∈ TPHR, the Riemann
curvature tensor only works if the tangent space of the manifold is equipped with the
additional structure of a Hilbert space (or an indefinite Hilbert space, as in general
relativity). However, the notion of curvature in the sense of Cartan is independent of
such an additional structure. In terms of mathematics, the notion of curvature can
be introduced without using a metric, one only needs what is called a connection.
In terms of physics, this corresponds to the transport of information.

5.11 Ariadne’s Thread in Gauge Theory
In terms of physics, roughly speaking, gauge theory describes additional
internal degrees of freedom of physical systems which do not affect the
physics.
For example, the choice of the potential does not influence the forces in classical
Newtonian mechanics.
From the mathematical point of view, gauge theory studies invariants under
gauge transformations.
Only such gauge invariants can be observed in physical experiments. Therefore,
gauge theory is part of the theory of invariants. In order to explain the basic ideas
of gauge theory, let us return to the hyperbolic plane HR.

5.11.1 Parallel Transport of Physical Information – the Key to
Modern Physics
In modern physics, the Huygens principle is replaced by the parallel transport
of physical information.
Folklore

5.11.2 The Phase Equation and Fiber Bundles
In a natural way, the global mathematical description of physical fields is
based on the language of bundles.
Folklore
The main trick due to ´Elie Cartan is to reduce the parallel transport of a physical
field to a dynamical system in a principal fiber bundle (Cartan’s method of moving
frame). To this end, we will replace the differential equation of parallel transport
by the phase equation.

Summarizing, we get the following:
The global curvature form F (defined on the principal fiber bundle P =
HR × GL(2,R) with values in the Lie algebra gl(2,R)) represents an invariant
mathematical object which carries all the information on the gauge
transformations of the Cartan curvature 2-form on the base manifold HR.

5.11.4 Perspectives
The approach above was chosen in order to display the historical development from
Gauss’ theorema egregium to Cartan’s structural equation. In modern differential
geometry, the axiomatic presentation reverses the historical order.
• The starting point is the fundamental phase equation, as a differential equation
on a principal fiber bundle P.
• More precisely, one starts with a velocity field on P which splits into horizontal
and vertical velocity vectors, with respect to the fibers.
• This velocity vector field generates the dynamical system of parallel transport.
The corresponding ordinary differential equation coincides with the equation of
parallel transport, which is based on the connection 1-form A on P.
• Projection onto horizontal vector fields is used in order to replace the differential
dω by the covariant differential Dω of differential forms ω on the principal fiber
bundle P.
• This yields the curvature form
F := DA, (5.112)
and the Bianchi identity DF = 0. It turns out that (5.112) is equivalent to
Cartan’s structural equation (5.110).
The extremely elegant formula (5.112) generalizes both the theorema egregium
of Gauss and the relation between the electromagnetic field tensor F and the 4-
potential A in Maxwell’s theory of electromagnetism. Moreover, formula (5.112) lies
at the heart of both the Standard Model in elementary physics and Einstein’s theory
of general relativity. In terms of mathematics, this approach effectively describes
all kinds of differential geometries:
• The typical Lie group of the principal fiber bundle P is the symmetry group of
the geometry under consideration.
• Physicists are interested in the study of partial differential equations for physical
fields. In this general setting, physical fields ψ are sections of a vector bundle V
associated to the principal fiber bundle P.
• There exists a natural way of transplanting the parallel transport from P to V.
• The parallel transport on V generates the corresponding covariant differentiation
for physical fields, which is used in order to formulate the basic partial differential
equations for the physical field under consideration.
The point is that one has to replace product bundles by general fiber bundles.
Observe that:
General fiber bundles are obtained by gluing local product bundles together,
with the aid of a cocycle.
In particular, the choice of arbitrary Lie groups allows us to take all kind of symmetries
into account. In 1872 Felix Klein (1849–1925) formulated his Erlangen
program:
Geometry is the invariant theory of transformation groups (symmetry
groups).
Sophus Lie (1842–1899), ´Elie Cartan (1859–1951), and their successors realized
this program in differential geometry by investigating Lie groups and the invariant
calculus of differential forms. This is a fascinating chapter in the history of mathematics
and physics. We will thoroughly study this in Vol. III. We also refer to
the standard textbook by S. Kobayashi and K. Nomizu, Foundations of Differential
Geometry, Vols. 1, 2, Wiley, New York, 1963.

5.12 Classification of Two-Dimensional Compact
Manifolds
The notion of manifold is of fundamental importance for both mathematics
and physics. There arises the problem of classifying manifolds in terms
of topology. This has been one of the most important research topics in
topology in the last 150 years.
Folklore
In 1863 M¨obius classified the compact orientable 2-manifolds.
In 1865 M¨obius published a strange su***ce called the
M¨obius strip nowadays. This twisted su***ce is obtained from a rectangle by gluing
together two opposite sides in a twisted manner (Fig. 5.26). If we walk along this
su***ce, then we reach both sides of the rectangle. Therefore, the M¨obius strip is a
one-sided su***ce, and hence it is not possible to define an orientation. Intuitively,
a su***ce is oriented iff the movement of a small oriented circle along a closed
curve never changes the orientation of the circle. The main theorem on classical
topological su***ce theory reads as follows:
The compact 2-manifolds M and N (without or with boundary) are homeomorphic
iff the following three conditions hold:
(i) M and N have the same genus g,
(ii) M and N have the same number of boundaries, and
(iii) both M and N are either orientable or non-orientable.
The genus attains the values g = 0, 1, 2, . . . Let us first consider a few typical
examples. Recall that the Euler characteristic of the 2-manifold M is given by
χ(M) = β0 − β1 + β2,
where β0, β1, β2 are the Betti numbers of M. Since the 2-manifold M is arcwise
connected, we always have β0 = 1. Moreover, we have β2 = 1 (resp. β2 = 0) iff M
is orientable (resp. non-orientable).
• The 2-sphere S2r
of radius r > 0 : This su***ce has the genus g = 0, the Betti
numbers
β0 = β2 = 1, β1 = 0
and the Euler characteristic χ(S2r
) = 2. The fundamental group is trivial,
π1(S2r
) = 0, that is, the sphere S2r
is simply connected.
• The 2-dimensional torus T2 : This su***ce can be obtained by identifying the
opposite boundary points of a rectangle (Fig. 5.27). Alternatively, the torus is
homeomorphic to a sphere with one handle attached to it (Fig. 5.25(a)). This
su***ce has the genus g = 1, the Betti numbers
β0 = β2 = 2, β1 = 2,
and the Euler characteristic χ(T2) = 0. For the additive fundamental group, we
get
π1(T2) = Z ⊕ Z.
This reflects the fact that there exist two different types of closed curves on the
torus which cannot be contracted to one point. For example, in Fig. 5.27(c) this
concerns the equator ABA and the meridian circle PAP. Consequently, the torus
T2 is not simply connected.
• The real 2-dimensional projective space P2 : This space is obtained from the
closed unit disc by identifying diametrically opposed boundary points of the unit
disc with each other (Fig. 5.28). The topological space P2 is compact, arcwise
connected, non-orientable, and it has the Betti numbers
β0 = 1, β1 = β2 = 0.
This yields the Euler characteristic χ(P2) = 1. The genus of P2 is given by g = 1,
as we will discuss below. The additive fundamental group of P2 is given by
π1(P2) = Z2.
This group consists of the two elements {0, 1} with 1+1 = 0. The element ”1” of
π1(P2) corresponds to the boundary curve PQP . This curve cannot be
continuously contracted to a point within P2. Thus, the projective space P2 is not
simply connected. In contrast to this, taking the identification of diametrically
opposed boundary points into account, the curve PQPQP can be continuously
contracted to the center of the unit disc. This corresponds to ”1+1=0.”
Let us now study the general case. We are given the compact 2-manifold M
of genus g. Let r = 0, 1, 2, . . . be the number of boundaries. Then there exists a
manifold N which is homeomorphic toMand which represents one of the following
normal forms:76
(I) Suppose that M is orientable.
(I-1) Let r = 0. Then the normal form N is obtained from the unit sphere S2 by
taking 2g open discs away and by attaching g handles. The genus is also equal
to the number of ‘holes’ (Fig. 5.25).
• Betti numbers of M: β0 = β2 = 1, β1 = 2g.
• Euler characteristic of M: χ = 2− 2g.
• Additive fundamental group of M: π1 = Z ⊕ . . . ⊕ Z (2g summands).
(I-2) Let r > 0. The normal form N is obtained from (I-1) by taking r open discs
away .
• Betti numbers of M: β0 = β2 = 1 and β1 = 2g + r − 1.
• Euler characteristic of M: χ = 3− 2g − r.
(II) Suppose that M is non-orientable.
(II-1) Let r = 0. The normal form N is obtained from the unit sphere S2 by taking
g open discs away, and by identifying diametral points of the boundary circles
with each other. The number g is the genus of M. For example, the su***ce
pictured in Fig. 5.30(a) is homeomorphic to the projective space P2.
• Betti numbers of M: β0 = 1, β1 = g − 1, β2 = 0. Here, g = 1, 2, . . .
• Euler characteristic of M: χ = 2− g.
(II-2) Let r > 0. The normal form is obtained from (II-1) by taking r open discs
away. Again the number g is called the genus of M.
• Betti numbers of M: β1 = 1, β1 = g + r − 1, β2 = 0.
• Euler characteristic of M: χ = 2− g − r.
In particular, the M¨obius strip corresponds to g = 1 and r = 1. Hence β1 = 1
and χ = 0.

5.13 The Poincar´e Conjecture and the Ricci Flow
The study of mathematics, like the Nil, begins in minuteness, but ends in
magnificence.
C.C. Colton, 1820
Friedman’s 1982 proof of the 4-dimensional Poincar´e hypothesis was an
extraordinary tour de force. His methods were so sharp that as to actually
provide a complete classification of all compact simply connected topological
4-manifolds, yielding many previously unknown examples of such
manifolds.
John Milnor, 1986
The Poincar´e conjecture for 3-manifolds was one of the seven Millenium Prize Problems
announced by the Clay Mathematics Institute in Cambridge, Massachusetts,
in the year 2000 (see Sect. 1.7 of Vol. I).
The topological characterization of the 2-sphere. The following result
was already known in the second half of the 19th century:
A compact simply connected 2-manifold is homeomorphic to
a 2-sphere.
This homeomorphism can be chosen as a diffeomorphism.
The topological characterization of the 3-sphere. The famous Poincar´e
conjecture claims the following:
A compact simply connected 3-manifold is diffeomorphic to
a 3-sphere.
This conjecture was proven by Grigori Perelman (born 1966) in 2003. He invented
an ingenious approach for solving this outstanding problem. The main idea comes
from physics. Here, Perelman uses the physical picture of the flow of fluid particles
in order to deform the original 3-manifold into the final 3-sphere. More precisely,
he applies the so-called Ricci flow on manifolds which was thoroughly studied in
the pioneering papers by Richard Hamilton in the 1980s and 1990s. The Ricci flow
is governed by the partial differential equation
∂g(t)/∂t= −2Ric(g(t)), t≥ 0. (5.113)
Here, the parameter t can be regarded as time. This equation describes the timedeformation
of the metric tensor g of a Riemannian manifold M. Equation (5.113)
is of the type of a diffusion process (or heat conduction process). The main difficulty
is that there may appear singularities during the time evolution. One has to control
the possible types of singularities, and one has to regularize and renormalize them
(see the hints for further reading on page 351).
The generalization of the idea of the flow of fluid particles is also crucial for
quantum physics. In Sect. 7.11.5 we will show that the Feynman path integral
corresponds to a diffusion process in imaginary time (the Feynman–Kac formula).
Moreover, the modern approach to renormalization in quantum field theory is based
on the flow generated by the renormalization group (see Chap. 3 of Vol. I).
The homotopy type of a topological space. Recall from Vol. I the following
terminology. Let X and Y be topological spaces. Let idX denote the identity map
on X. The two continuous maps f, g : X → X are called homotopic iff there exists
a continuous map H : X × [0, 1] → X such that
H(x, 0) = f(x) and H(x, 1) = g(x) for all x ∈ X.
We write f ∼ g. The map H is called a homotopy.
By definition, the space X has the same homotopy type as the space Y iff there
exist continuous maps F : X → Y and G : Y → X such that
GF ∼ idX and FG ∼ idY .
That is, the composite maps GF : X → X and FG : Y → Y are homotopic to the
corresponding identity maps. We also say that the space X is homotopy equivalent
to the space Y . For example, a topological space is called contractible iff it has the
same homotopy type as a single point (see Sect. 5.5 of Vol. I)..
By definition, the topological space X has the same topological type as the
topological space Y iff X is homeomorphic to Y . Explicitly, this means that there
exists a bijective continuous map F : X → Y such that the inverse map F−1 : Y →
X is also continuous. Setting G := F−1, we get GF = idX and FG = idY. Thus, if
X and Y have the same topological type, then they also have the same homotopy
type. However, the converse is not always true.
The topological characterization of the n-sphere. The generalized Poincar
´e conjecture reads as follows:
If an n-manifold has the same homotopy type as an n-sphere, then it has
actually the same topological type as an n-sphere.
Nowadays we know that this statement is true for all dimensions n = 1, 2, . . .
For n ≥ 5, the proof was given by Smale, and independently by Stallings and
Zeeman and by Wallace in 1960-61. For n = 4, Freedman gave the proof in 1982.
Perelman settled the most difficult case n = 3 in 2002. For their seminal results,
Smale, Freedman and Perelman were awarded the Fields medal in 1966, 1986, and
2006, respectively. Perelman refused the award. Another formulation of the Poincar´e
conjecture reads as follows:
If an n-manifold has the same fundamental group and the same
homology as the n-sphere, then it is actually homeomorphic to the
n-sphere.78
This is true for all dimensions n = 1, 2, . . . The positive answer to the Poincar´e
conjecture tells us the following highly nontrivial result:
Algebraic topology is able to detect spheres in all dimensions.

5.14 A Glance at Modern Optimization Theory
In the 1950s, modern control theory was founded by generalizing the duality between
wave fronts and light rays.
(i) Dynamic programming (generalized wave fronts): Bellman created dynamic programming
by generalizing the Hamilton–Jacobi partial differential equation to
the Bellman functional equation. In geometrical optics, the eikonal function S
measures the time needed for the propagation of light. In dynamic programming,
the function S measures the quantity to be optimized (e.g., the costs of
a production process).
(ii) Pontryagin’s maximum principle (generalized light rays): Pontryagin generalized
the Hamilton canonical equations for light rays (and the maximum principle
in geometrical optics) to the computation of optimal trajectories in modern
technology. Let us discuss some important examples.
• For the return of a spaceship from moon to earth, one has to compute a
trajectory such that the heating of the spaceship remains minimal. Of interest
in the optimal solution is the fact that the spaceship penetrates the
earth’s atmosphere rather deeply (from 120 km altitude to 50 km) and then
it climbs again to the altitude of 75 km. On the other hand, the velocity falls
almost monotonically. The computation (performed by Roland Bulirsch for
the NASA in the 1960s) can be found in Stoer and Bulirsch, Introduction
to Numerical Analysis, Springer, New York, 1993.
• For the moon landing, one needs a feed-back control program which guarantees
minimal fuel consumption of the moon landing ferry. Here, the braking
process is controlled by measuring the distance between the ferry and the
moon su***ce.
• For the flight to Mars, one needs a trajectory of the spaceship which again
guarantees minimal fuel consumption, by taking the gravitational forces of
the planets in our solar system into account.
The proof for the validity of Pontryagin’s maximum principle is highly sophisticated.
For a detailed study of (i) and (ii) in the setting of nonlinear functional
analysis, we refer to Zeidler (1986), Vol. III, quoted on page 353.

6. The Principle of Critical Action and the
Harmonic Oscillator – Ariadne’s Thread in
Classical Mechanics
Since the divine plan is the most perfect thing there is, there can be no
doubt that all actions in the universe can be determined by the calculus
of the minima and maxima from the corresponding causes.
Leonhard Euler
The history of the principle of least action has often been described. Yet
the matter is still controversial, and there seems to be no general agreement
who invented the principle, Leibniz (1646–1717), Euler (1707–1783),
or Maupertuis (1698–1759). . . We mention that the first mathematical
treatment of the action principle was given by Euler in the Additamentum
of his Methodus inveniendi.
Mariano Giaquinta and Stefan Hildebrandt, 1996
By generalizing the method of Euler in the calculus of variations, Lagrange
(1736–1813) discovered, how one can write, in a single line, the basic equation
for all problems in analytic mechanics.
Carl Gustav Jakob Jacobi (1804–1851)
When we quantize a classical theory, wave packets behave like particles. . .
A wave packet might decay into two wave packets. When two wave packets
come near to each other, they scatter and perhaps produce more wave
packets. This naturally suggests the physics of particles can be described
in these terms. . .
Quantum field theory grew out of essentially these sorts of physical ideas.
It struck me as limiting that even after some 75 years, the whole subject
of quantum field theory remains rooted in this harmonic paradigm, to use
a dreadfully pretentious word. We have not been able to get away from
the basic notions of oscillations and wave packets. Indeed, string theory,
the heir to quantum field theory, is still firmly founded on this harmonic
The aim of this and the following chapter is to explain the basic physical and
mathematical ideas of classical mechanics and quantum mechanics by considering
the so-called harmonic oscillator. In all fields of physics, one encounters oscillating
systems. Let us mention the following examples:
• electromagnetic waves and light (photons);
• laser beams (coherent states);
• oscillating molecules in a gas or a liquid;
• sound waves (phonons);
• oscillations of a crystal lattice (phonons);
• oscillations of a string (e.g., a violin string);
• waves in a plasma (plasmons);
• matter waves of elementary particles (e.g., electrons);
• gravitational waves (gravitons).
The harmonic oscillator represents the simplest oscillating system. The quantization
of the harmonic oscillator is the basis of quantum mechanics, quantum field theory,
and condensed matter physics.
System of physical units. In this chapter, we will use the international system
of units, SI (see the Appendix of Vol. I).

6.1 Prototypes of Extremal Problems
The calculus of variations has its roots in extremal problems for real-valued
functions.
Folklore
The one-dimensional problem. Let f : J → R be a smooth function on the
open interval J. Consider the minimum problem
f(x) = min!, x∈ J. (6.1)
Let us recall some standard results from classical calculus.
(i) Necessary condition for a local minimum: If x0 is a solution of (6.1), then
f'(x0) = 0 and f''(x0) ≥ 0.
(ii) Sufficient condition for a local minimum: If f'(x0) = 0 and f''(x0) > 0, then
the function f has a local minimum at the point x0. This means that there
exists a sufficiently small positive number ε such that f(x) ≥ f(x0) for all
x ∈]x0 − ε, x0 + ε[.
(iii) Sufficient condition for a global minimum: If f'(x0) = 0, f''(x0) > 0, and
f''(x) ≥ 0 on J, then the function f is convex on J, and the minimum problem
(6.1) has the unique solution x0.
Now consider the more general problem
f(x) = critical!, x0 ∈ J. (6.2)
By definition, the point x0 is a solution of (6.2) iff f'(x0) = 0. We say that x0 is
a critical point of f, and the function is critical (or stationary) at the point x0.
Intuitively, the function f is critical at the point x0 iff the tangent line of the graph
of the function f at the point x0 is horizontal (Fig. 6.1). For example, the function
f : R → R given by f(x) := x2 has a global minimum at the point x0 = 0. In
contrast to this, the function f(x) := x3 has the unique critical point x0 = 0,
but no minimal point. The function f(x) := sinx has precisely the critical points
x0 = ±π(n + 1
2) with n = 0, 1, 2, . . .

6.2 The Motion of a Particle

6.3 Newtonian Mechanics
The rise of modern science was accompanied with the replacement of authorities
or traditions by causes in explaining phenomena. One of the ultimate
goals of science is to understand the world, and this is approached by
scientific explanation, that is, by finding out causes for various phenomena.
According to Aristotle, however, there are different kinds of cause:
material, formal, efficient, and final causes. Before the rise of modern science,
teleological explanation based on the notion of final cause was a
dominant mode of explanation. With the revival of Neoplatonism, Archimedianism
and atomism in the Renaissance, there began a transformation
in basic assumptions of scientific explanation. Copernicus, Kepler, Galileo,
and Descartes, for example, believed that the underlying truth and universal
harmony of the world can be perfectly represented by simple and exact
mathematical expressions. The mathematization of nature led to a certain
degree of popularity of formal cause. But the most popular and powerful
conception of causality, in fighting, against the teleological explanation,
was a mechanical one based on the notion of efficient cause. Different from
final and formal causes, the idea of efficient cause focuses on how the cause
is transmitted to the effect, that is, on the mode of transmission. According
to the classical mechanical view, causality can be reduced to the laws
of motion of bodies in space and time. . .
Tian Yu Cao, 1998
Conceptual Developments of 20th Century Field Theories
So that we may say now that the door is opened, for the first time, to a
new method fraught with numerous and wonderful results which in future
years will command the attention of other minds.
Galileo Galilei (1564–1642)
Lex prima: A stationary body will remain motionless, and a moving body
will continue to move in the same direction with unchanging speed unless
it is acted on by some force.
Lex secunda: The time-rate-of-change of the momentum of a body is proportional
to the force.
Lex tertia: If any body exerts a force on another object, then the second
object also exerts an equal and opposite force on the first.
It remains that, from the same principles, I now demonstrate the frame of
the System of the World.
Isaac Newton (1643–1727)
Philosophiae Naturalis Principia Mathematica, London, 16877
Who, by a vigor of mind almost divine, the motions and figures of the
planets, the paths of comets, and the tides of the sea first demonstrated.8
Newton’s Epitaph, Westminster Abbey, London
When one considers all that Newton achieved, and the cultural and scientific
environment in which he achieved it, there is reason to regard him as
the greatest scientist – and perhaps the greatest genius – that ever lived.
Anthony Philip French

6.4 A Glance at the History of the Calculus of
Variations
Bees – by virtue of a certain geometrical forethought – know that the
hexagon is greater than the square and the *** and will hold more
money for the same expenditure of material.
Pappus of Alexandria, 300 B.C.
Every process in nature will occur in the shortest possible way.
Leonardo da Vinci (1452–1519)
A light ray between two points needs the shortest possible time.
Pierre de Fermat (1601–1665)
Johann Bernoulli, professor of mathematics, greets the most sophisticated
mathematicians in the world. Experience shows that noble intellectuals
are driven to work for the pursuit of the knowledge by nothing more than
being confronted with difficult and useful problems.
Six months ago, in the June edition of the Leipzig Acta eruditorum (journal
of scientists), I presented such a problem. The allotted six-month deadline
has now gone by, but no trace of a solution has appeared. Only the famous
Leibniz informed me that he had unravelled the knot of this brilliant and
outstanding problem, and he kindly asked me to extend the deadline until
next Easter. I agreed to this honorable request. . . I will repeat the problem
here once more.
Two points, at different distances from the ground, and not in a vertical
line, should be connected by such a curve that a body under the influence
of gravitational forces passes in the shortest possible way from the upper
to the lower point .
Johann Bernoulli, January 1697
This paper solves my brother’s problem, to whom I will set other problems
in return.
Jakob Bernoulli, May 1697
The Euler Calculus of Variations (Methodus inveniendi) from the year
1744 is one of the most beautiful mathematical works that has ever been
written.
Constantin Carath´eodory (1873–1950)
Read Euler, he is the master of us all.
Marquise de Pierre Simon Laplace (1749–1824)
One needs to have delved but little into the principles of differential calculus
to know the method of how to determine the greatest and least
ordinates of curves. But there are maxima or minima problems of a higher
order, which in fact depend on the same method, which however cannot
be subjected to this method. These are the problems where it is a matter
of finding the curves themselves.
The first problem of this type, which the geometers solved, is that of
the brachistochrone or the curve of fastest fall which Johann Bernoulli
proposed toward the end of the preceding century. One attained this only
in special ways, and it was only some time later and on the occasion of the
investigations concerning isoperimetric problems that the great geometer
of whom we just spoke and his extraordinary brother Jakob Bernoulli gave
some rules in order to solve several other problems of this type.
But since these rules were not of sufficient generality, the famous Euler
undertook to refer all investigations of this type to a general method.  But
even as sophisticated and fruitful as his method is, one must nevertheless
confess that it is not sufficiently simple. . . Now here one finds a method
which requires only a simple use of the principles of differential and integral
calculus.
Joseph Louis Lagrange, 1762
As I see, your analytic solution of the isoperimetric problem contains all
that one can wish for in this situation. I am very happy that this theory
which I have treated since the first attempts almost alone, has been brought
precisely by you to the highest degree of perfection.
The importance of the situation has occasioned me with the help of your
new insights to myself conceive of an analytic solution, but which I shall
not make known before you have published your deliberations, in order
not to deprive you of the least part of the fame due you.
Euler, in a letter to the young Lagrange

6.5 Lagrangian Mechanics
The mathematician is perfect only in so far as he is a perfect being, in so
far as he perceives the beauty of truth; only then will his work be thorough,
transparent, comprehensive, pure, clear, attractive, and even elegant. All
this is necessary in order to resemble Lagrange.
Johann Wolfgang von Goethe (1749–1832)
Wilhelm Meisters Wanderjahre

Jacobi’s Accessory Eigenvalue Problem
Returning to the concepts of maximum and minimum, it is a nuisance that
there reigns such confusion in these words. One says that an expression
attains a maximum or a minimum if one simply wishes to say that it is
critical (or extremal) and hence its first variation vanishes, also in the case
when neither a minimum nor a maximum occurs.
Carl Gustav Jacobi (1804–1851)

6.5.4 The Morse Index
The Morse index describes the global behavior of the action functional of
the harmonic oscillator with respect to arbitrary time intervals. This global
behavior is governed by the appearance of focal points of the trajectories
of the harmonic oscillator, which correspond to focal points in geometric
optics.
Folklore

6.6 Symmetry and Conservation Laws
Newton and his successors noticed that there exist conservation laws that simplify
the integration of the equations of motion. For example, this concerns the
conservation of the following quantities: energy, momentum, angular momentum,
Runge–Lenz vector. In 1918, Emmy Noether (1882–1935) proved a general theorem
which shows that
The symmetries of the Lagrangian are responsible for conservation laws.
For example, invariance of the Lagrangian under time translations leads to conservation
of energy. More general, we will show in Sec. 6.6.2 that smooth continuous
symmetries of the action integral imply conservation laws.

6.7 The Pendulum and Dynamical Systems
Henri Poincar´e (1854–1912) originated not only new theories, but completely
new branches of mathematics like the theory of dynamical systems,
differential topology, and algebraic topology. His ideas are so great,
his way of thinking of and looking at mathematical reality has been so
widely accepted that to us, his descendants, it seems strange that people
have thought differently, for example, that dynamical systems should be
considered on manifolds, and not only on Rn – because indeed a plane
pendulum is a motion on the circle, and a spherical pendulum is a motion
on a sphere.
Krysztof Maurin, 1999
Motions of mass point systems are frequently governed by constraints (e.g., the
oscillations of molecules). Whereas the free motion of N mass points in Euclidean
space has 3N degrees of freedom, constrained motions possess less than 3N degrees
of freedom. The Lagrangian approach to mechanics allows an elegant reduction to
the true number of degrees of freedom by considering the motion with respect to
appropriately chosen local coordinates. This corresponds to the theory of dynamical
systems on manifolds. In order to illustrate this, let us consider the motion x = x(t)
of a pendulum of mass m and length  l.
In a right-handed Cartesian coordinate system, we set x = xi + yj + zk (Fig.
6.9). The gravitational force
F = −mgk
acts on a particle of mass m. Here, g = 9.81m/s2 is the acceleration of gravity.
Letting U(x) := mgz, we get F = −grad U. Therefore, the force F has the
potential U.

6.7.3 The Phase Space of the Pendulum and Bundles
It was a great achievement of Poincar´e (1854–1912) to show that the proper
domain of analytic dynamics is the cotangent bundle TMd of the position
space M – this discovery was so fundamental that nowadays it seems to
be natural and obvious.
Krysztof Maurin, 1996
The categories of differentiable manifolds and vector bundles provide a
useful context for the mathematics needed in mechanics, especially the
new topological and qualitative results.
Ralph Abraham and Jerrold Marsden, 1978
Too often in the physical sciences, the space of states is postulated to be a
linear space when the basic problem is essentially nonlinear; this confuses
the mathematical development.
Stephen Smale, 1980
This section should help the reader to understand the intuitive roots of the language
of bundle theory in modern mathematics and physics. In mechanics, one has to
distinguish the following crucial notions:
• position space M;
• state space TM (tangent bundle of the manifold M);
• phase space TMd (cotangent bundle of M).
Let us discuss this for the prototype of a mechanical system with nontrivial topology
– the circular pendulum.
Our strategy is to introduce only such quantities which have a geometrical
meaning, that is, they are independent of the choice of local coordinates.

Perspective. For a general mechanical system of n degrees of freedom with
n = 1, 2, . . ., the following hold:
• the position space M is a real n-dimensional manifold,
• the state space TM (tangent bundle of M) is a 2n-dimensional manifold, and
• the phase space TMd (cotangent bundle) is a 2n-dimensional symplectic manifold.
The Lagrangian formulation of mechanics is based on the Lagrangian
L : TM → R,
whereas the dual Hamiltonian formulation of mechanics is based on the Hamiltonian
H : TMd → R,
which represents the energy function of the mechanical system, as a rule. In contrast
to the cotangent bundle TMd, the tangent bundle TM is not always a symplectic
manifold. Therefore, the Hamiltonian formulation of mechanics has advantages over
the Lagrangian formulation. In statistical mechanics, the Hamiltonian approach will
allow us to use the volume measure of the phase space in order to construct the key
probability measure (see Sect. 7.17.5 on page 645). Summarizing, the language of
manifolds allows us to study the global aspects of the motion of mechanical systems.
The complexity of a mechanical system is reflected by the complex topology
of the tangent bundle and the cotangent bundle.

6.8 Hamiltonian Mechanics
The Hamiltonian approach to mechanics is centered at the concept of energy. In
this setting, the following two important results are quite natural:
• Conservation of energy, and
• conservation of phase volume.
The latter property is crucial for classical statistical physics (Gibbs measure). From
the geometric point of view, the passage from Lagrangian mechanics to Hamiltonian
mechanics corresponds to a passage
• from the tangent bundle TM of the position space M (position, velocity vector)
• to the dual cotangent bundle TMd of M (position, differential form).
This transformation is called the Legendre transformation which is a contact transformation
in the sense of Lie. The cotangent bundle TMd always carries a natural
symplectic structure. Note the following:
• Lagrangian mechanics corresponds to Riemannian geometry (the metric is given
by the kinetic energy).
• Hamiltonian mechanics corresponds to symplectic geometry.
One of the great problems of mathematics and physics in the 19th century consisted
in solving the N-body problem in celestial mechanics.
The aim was to investigate the stability of our solar system.
To simplify considerations, Jacobi and his successors used transformations of time
and position which preserve the form of the canonical equations.
Such transformations are called canonical transformations.
It turns out that canonical transformations coincide with symplectic transformations.
That is, canonical transformations preserve the symplectic structure, and
hence they present the symmetry transformations of symplectic geometry. When
creating his mechanics, Hamilton (1805–1865) was motivated by an analogy between
mechanics and Huygens’ geometrical optics created in the 17th century:
• The trajectories q = q(t), p = p(t) of a particle in mechanics correspond to light
rays (solutions of the canonical ordinary differential equations), and
• the action function S = S(q, t) in mechanics corresponds to the eikonal function
in geometric optics, which determines the wave fronts (solutions of the Hamilton–
Jacobi partial differential equation).
• The duality between trajectories and wave fronts was fully established by
Carath´eodory in the framework of his ’royal road to the calculus of variations’
in 1925. This is based on the fundamental notion of geodesic fields.
The relation between geometric optics and wave optics plays a fundamental role
for understanding Schr¨odinger’s quantum mechanics which he discovered in 1926.
In the 1950s, optimal control theory was invented independently by Bellman and
Pontryagin. This theory allows many applications in technology (e.g., moon landing
and return of a spaceship to earth):
• Pontryagin’s theory generalizes Hamilton’s canonical equations (Pontryagin’s
maximum principle and light rays), whereas
• Bellman’s theory of dynamic programming generalizes the Hamilton–Jacobi partial
equation (the Hamilton–Jacobi–Bellman equation and wave fronts).
In the Bellman approach, the action function S corresponds to the cost function.
A detailed study can be found in Zeidler (1986), Vol. III (see the references on
page 1049). In what follows, let us study the main ideas on an elementary level by
considering the harmonic oscillator as a prototype.

6.9 Poissonian Mechanics
The Poissonian formulation of mechanics has the following two advantages:
• Conservation laws can be expressed in terms of Poisson brackets.
• Quantum mechanics can be obtained from classical mechanics by replacing the
Poisson bracket {A,B} with the commutator
1/ih· [A,B]−
where the Lie bracket [A,B]− is equal to AB−BA. This was implicitly discovered
by Heisenberg in 1925. The general quantization rule was formulated by Dirac
in 1926.

6.10 Symplectic Geometry
A deeper understanding of classical mechanics is based on symplectic geometry.
By definition, a quantity belongs to symplectic geometry iff it is invariant under
symplectic transformations. Let us explain the basic ideas.

6.11 The Spherical Pendulum
Two-dimensional spheres are the simplest curved su***ces. They serve as
prototypes for the geometry and analysis of manifolds.
Folklore

6.13.2 Lagrangian Submanifolds in Symplectic Geometry
In geometrical optics, one wants to construct wave fronts by means of families of
light rays. The point is that only special families of light rays allow this construction.
This is intimately related to the notion of Lagrange brackets, which were introduced
by Lagrange in the 18th century in order to simplify computations in celestial mechanics
in the framework of perturbation theory. The point is that the Lagrange
brackets of a family of light rays are constant in time along the light rays (i.e.,
they are first integrals of the Hamilton canonical equations). In modern symplectic
geometry, the Lagrange brackets are reformulated as Lagrangian submanifolds of
a symplectic manifold. Replacing light rays by trajectories of particles in classical
mechanics, we will study
• the construction of a solution of the Hamilton–Jacobi partial differential equation
• by the help of suitable families of trajectories which satisfy the Hamilton canonical
system of ordinary differential equations.

7. Quantization of the Harmonic Oscillator –
Ariadne’s Thread in Quantization
Whoever understands the quantization of the harmonic oscillator can understand
everything in quantum physics.
Folklore
Almost all of physics now relies upon quantum physics. This theory was
discovered around the beginning of this century. Since then, it has known
a progress with no analogue in the history of science, finally reaching a
status of universal applicability.
The radical novelty of quantum mechanics almost immediately brought a
conflict with the previously admitted corpus of classical physics, and this
went as far as rejecting the age-old representation of physical reality by
visual intuition and common sense. The abstract formalism of the theory
had almost no direct counterpart in the ordinary features around us, as,
for instance, nobody will ever see a wave function when looking at a car
or a chair. An ever-present randomness also came to contradict classical
determinism.
Roland Omn`es, 1994
Quantum mechanics deserves the interest of mathematicians not only because
it is a very important physical theory, which governs all microphysics,
that is, the physical phenomena at the microscopic scale of 10
−10m, but
also because it turned out to be at the root of important developments of
modern mathematics.
Franco Strocchi, 2005
In this chapter, we will study the following quantization methods:
• Heisenberg quantization (matrix mechanics; creation and annihilation operators),
• Schr¨odinger quantization (wave mechanics; the Schr¨odinger partial differential
equation),
• Feynman quantization (integral representation of the wave function by means of
the propagator kernel, the formal Feynman path integral, the rigorous infinitedimensional
Gaussian integral, and the rigorous Wiener path integral),
• Weyl quantization (deformation of Poisson structures),
• Weyl quantization functor from symplectic linear spaces to C
∗
-algebras,
• Bargmann quantization (holomorphic quantization),
• supersymmetric quantization (fermions and bosons).
We will choose the presentation of the material in such a way that the
reader is well prepared for the generalizations to quantum field theory to
be considered later on.
Formally self-adjoint operators. The operator A : D(A) → X on the complex
Hilbert space X is called formally self-adjoint iff the operator is linear, the domain
of definition D(A) is a linear dense subspace of the Hilbert space X, and we have
the symmetry condition
<χ|Aϕ >=< Aχ|ϕ> for all χ, ψ ∈ D(A).
Formally self-adjoint operators are also called symmetric operators. The following
two observations are crucial for quantum mechanics:
• If the complex number λ is an eigenvalue of A, that is, there exists a nonzero
element ϕ ∈ D(A) such that Aϕ = λϕ, then λ is a real number. This follows
from λ =< ϕ|Aϕ> =< Aϕ|ϕ >= λ†.
• If λ1 and λ2 are two different eigenvalues of the operator A with eigenvectors ϕ1
and ϕ2, then ϕ1 is orthogonal to ϕ2. This follows from
(λ1 − λ2)<ϕ1|ϕ2 >= <Aϕ1|ϕ2> − <ϕ1|Aϕ2> = 0.
In quantum mechanics, formally self-adjoint operators represent formal observables.
For a deeper mathematical analysis, we need self-adjoint operators, which
are called observables in quantum mechanics.
Each self-adjoint operator is formally self-adjoint. But, the converse is not true. For
the convenience of the reader, on page 683 we summarize basic material from functional
analysis which will be frequently encountered in this chapter. This concerns
the following notions: formally adjoint operator, adjoint operator, self-adjoint operator,
essentially self-adjoint operator, closed operator, and the closure of a formally
self-adjoint operator. The reader, who is not familiar with this material, should
have a look at page 683. Observe that, as a rule, in the physics literature one does
not distinguish between formally self-adjoint operators and self-adjoint operators.
Peter Lax writes:
The theory of self-adjoint operators was created by John von Neumann to
fashion a framework for quantum mechanics. The operators in Schr¨odinger’s
theory from 1926 that are associated with atoms and molecules
are partial differential operators whose coefficients are singular at certain
points; these singularities correspond to the unbounded growth of the force
between two electrons that approach each other. . . I recall in the summer
of 1951 the excitement and elation of von Neumann when he learned that
Kato (born 1917) has proved the self-adjointness of the Schr¨odinger operator
associated with the helium atom.
And what do the physicists think of these matters? In the 1960s Friedrichs5
met Heisenberg and used the occasion to express to him the deep gratitude
of the community of mathematicians for having created quantum mechanics,
which gave birth to the beautiful theory of operators in Hilbert space.
Heisenberg allowed that this was so; Friedrichs then added that the mathematicians
have, in some measure, returned the favor. Heisenberg looked
noncommittal, so Friedrichs pointed out that it was a mathematician, von
Neumann, who clarified the difference between a self-adjoint operator and
one that is merely symmetric.“What’s the difference,” said Heisenberg.
As a rule of thumb, a formally self-adjoint (also called symmetric) differential operator
can be extended to a self-adjoint operator if we add appropriate boundary
conditions. The situation is not dramatic for physicists, since physics dictates the
‘right’ boundary conditions in regular situations. However, one has to be careful.
In Problem 7.19, we will consider a formally self-adjoint differential operator which
cannot be extended to a self-adjoint operator.
The point is that self-adjoint operators possess a spectral family which allows
us to construct both the probability measure for physical observables
and the functions of observables (e.g., the propagator for the quantum dynamics).
In general terms, this is not possible for merely formally self-adjoint operators.
The following proposition displays the difference between formally self-adjoint and
self-adjoint operators.
Proposition 7.1 The linear, densely defined operator A : D(A) → X on the complex
Hilbert space X is self-adjoint iff it is formally self-adjoint and it always follows
from
<ψ|Aϕ >=< χ|ϕ>
for fixed ψ, χ ∈ X and all ϕ ∈ D(A) that ψ ∈ D(A).
Therefore, the domain of definition D(A) of the operator A plays a critical role.

7.1 Complete Orthonormal Systems
A complete orthonormal system of eigenstates of an observable (e.g., the
energy operator) cannot be extended to a larger orthonormal system of
eigenstates.
Folklore
Basic question. Let H : D(H) → X be a formally self-adjoint operator on the
infinite-dimensional separable complex Hilbert space X. Physicists have invented
algebraic methods for computing eigensolutions of the form
Hϕn = Enϕn, n= 0, 1, 2, . . . (7.1)
The idea is to apply so-called ladder operators which are based on the use of commutation
relations (related to Lie algebras or super Lie algebras).We will encounter
this method several times. In terms of physics, the operator H describes the energy
of the quantum system under consideration. Here, the real numbers E0,E1,E2, . . .
are the energy values, and ϕ0, ϕ1, ϕ2, . . . are the corresponding energy eigenstates.
Suppose that ϕ0, ϕ1, ϕ2, . . . is an orthonormal system, that is,
<ϕk|ϕn> = δkn, k,n= 0, 1, 2, . . .
There arises the following crucial question.

7.2 Bosonic Creation and Annihilation Operators
Whoever understands creation and annihilation operators can understand
everything in quantum physics.
Folklore

7.3 Heisenberg’s Quantum Mechanics
Quantum mechanics was born on December 14, 1900, when Max Planck
delivered his famous lecture before the German Physical Society in Berlin
which was printed afterwards under the title “On the law of energy distribution
in the normal spectrum.” In this paper, Planck assumed that the
emission and absorption of radiation always takes place in discrete portions
of energy or energy quanta hν, where ν is the frequency of the emitted or
absorbed radiation. Starting with this assumption, Planck arrived at his
famous formula

for the energy density  of black-body radiation at temperature T.
Barthel Leendert van der Waerden, 1967
The present paper seeks to establish a basis for theoretical quantum mechanics
founded exclusively upon relationships between quantities which
in principle are observable.
Werner Heisenberg, 1925
The recently published theoretical approach of Heisenberg is here developed
into a systematic theory of quantum mechanics with the aid of mathematical
matrix theory. After a brief survey of the latter, the mechanical
equations of motions are derived from a variational principle and it is
shown that using Heisenberg’s quantum condition, the principle of energy
conservation and Bohr’s frequency condition follow from the mechanical
equations. Using the anharmonic oscillator as example, the question of
uniqueness of the solution and of the significance of the phases of the
partial vibrations is raised. The paper concludes with an attempt to incorporate
electromagnetic field laws into the new theory.
Max Born and Pascal Jordan, 1925
There exist three different, but equivalent approaches to quantum mechanics,
namely,
(i) Heisenberg’s particle quantization from the year 1925 and its refinement by
Born, Dirac, and Jordan in 1926,
(ii) Schr¨odinger’s wave quantization from 1926, and
(iii) Feynman’s statistics over classical paths via path integral from 1942.
In what follows we will thoroughly discuss these three approaches in terms of the
harmonic oscillator. Let us start with (i).

7.3.1 Heisenberg’s Equation of Motion
In a recent paper, Heisenberg puts forward a new theory which suggests
that it is not the equations of classical mechanics that are in any way at
fault, but that the mathematical operations by which physical results are
deduced from them require modification. All the information supplied by
the classical theory can thus be made use of in the new theory . . . We make
the fundamental assumption that the difference between the Heisenberg
products is equal to i times their Poison bracket
xy − yx = i{x, y}. (7.18)
It seems reasonable to take (7.18) as constituting the general quantum
conditions.
Paul Dirac, 1925
The general quantization principle.We are looking for a simple principle which
allows us to pass from classical mechanics to quantum mechanics. This principle
reads as follows:
• position q(t) and momentum p(t) of the particle at time t become operators,
• and Poisson brackets are replaced by Lie brackets,
{A(q, p),B(q, p)} ⇒ 1/ih[A(q, p),B(q, p)]−.
Recall that [A,B]− := AB − BA. Using this quantization principle, the classical
equation of motion (7.17) passes over to the equation of motion for the quantum
harmonic oscillator
ihq˙(t) = [q(t),H(q(t), p(t))]−,
ihp˙(t) = [p(t),H(q(t), p(t))]−                                                            (7.19)
together with
[q(t), p(t)]− = ihI. (7.20)
The latter equation is called the Heisenberg–Born–Jordan commutation relation.

The famous Heisenberg uncertainty inequality for the quantum harmonic oscillator
tells us that the state ϕn has the sharp energy En, but it is impossible
to measure sharply both position and momentum of the quantum particle at the
same time. Thus, there exists a substantial difference between classical particles
and quantum particles.
It is impossible to speak of the trajectory of a quantum particle.

7.3.3 Quantization of Energy
I have the best of reasons for being an admirer of Werner Heisenberg.
He and I were young research students at the same time, about the same
age, working on the same problem. Heisenberg succeeded where I failed. . .
Heisenberg - a graduate student of Sommerfeld - was working from the
experimental basis, using the results of spectroscopy, which by 1925 had
accumulated an enormous amount of data20. . .
Paul Dirac, 1968
The measured spectrum of an atom or a molecule is characterized by two quantities,
namely,
• the wave length λnm of the emitted spectral lines (where n,m = 0, 1, 2, . . . with
n > m), and
• the intensity of the spectral lines.
In Bohr’s and Sommerfeld’s semi-classical approach to the spectra of atoms and
molecules from the years 1913 and 1916, respectively, the spectral lines correspond
to photons which are emitted by jumps of an electron from one orbit of the atom or
molecule to another orbit. If E0 < E1 < E2 < .. . are the (discrete) energies of the
electron corresponding to the different orbits, then a jump of the electron from the
higher energy level En to the lower energy level Em produces the emission of one
photon of energy En − Em. According to Einstein’s light quanta hypothesis from
1905, this yields the frequency
νnm =(En − Em)/h, n>m (7.26)
of the emitted photon, and hence the wave length λnm = c/νnm of the corresponding
spectral line is obtained. The intensity of the spectral lines depends on the transition
probabilities for the jumps of the electrons. In 1925 it was Heisenberg’s philosophy
to base his new quantum mechanics only on quantities which can be measured in
physical experiments, namely,
• the energies E0,E1, . . . of bound states and
• the transition probabilities for changing bound states.

7.3.5 The Wightman Functions
Both the Wightman functions and the correlation functions of the quantized
harmonic oscillator are the prototypes of general constructions used
in quantum field theory.
Folklore

We define the n-point
Wightman function of the quantized harmonic oscillator by setting
Wn(t1, t2, . . . , tn) := 0|q(t1)q(t2) · · · q(tn)|0 (7.32)
for all times t1, t2, . . . , tn ∈ R. This is the vacuum expectation value of the operator
product q(t1)q(t2) · · · q(tn). In contrast to the operator function (7.31), the
Wightman functions are classical complex-valued functions. It turns out that
The Wightman functions know all about the quantized harmonic oscillator.
Using the Wightman functions, we avoid the use of operator theory in Hilbert space.
This is the main idea behind the introduction of the Wightman functions.

Perspectives. In 1956 Wightman showed that it is possible to base quantum
field theory on the investigation of the vacuum expectation values of the products of
quantum fields. These vacuum expectation values are called Wightman functions.
The crucial point is that the Wightman functions are highly singular objects in
quantum field theory. In fact, they are generalized functions. However, they are
also boundary values of holomorphic functions of several complex variables. This
simplifies the mathematical theory. Using a similar construction as in the proof
of the Gelfand–Naimark–Segal (GNS) representation theorem for C∗-algebras in
Hilbert spaces, Wightman proved a reconstruction theorem which shows that the
quantum field (as a Hilbert-space valued distribution) can be reconstructed from
its Wightman distributions.

7.3.6 The Correlation Functions
In contrast to the Wightman functions, the correlation functions reflect
causality.
Folklore
Parallel to (7.32), we now define the n-point correlation function (also called the
n-point Green’s function) by setting
Cn(t1, t2, . . . , tn) :=< 0|T (q(t1)q(t2) · · · q(tn))|0>        (7.37)
for all times t1, t2, . . . , tn ∈ R. Here, the symbol T denotes the time-ordering operator,
that is, we define
T (q(t1)q(t2) · · · q(tn)) := q(tπ(1))q(tπ(2)) · · · q(tπ(n))
where the permutation π of the indices 1, 2, . . . , n is chosen in such a way that
tπ(1) ≥ tπ(2) ≥ . . . ≥ tπ(n).

7.4 Schr¨odinger’s Quantum Mechanics
In particular, I would like to mention that I was mainly inspired by the
thoughtful dissertation of Mr. Louis de Broglie (Paris, 1924). The main
difference here lies in the following. De Broglie thinks of travelling waves,
while, in the case of the atom, we are led to standing waves. . . I am most
thankful to Hermann Weyl with regard to the mathematical treatment of
the equation of the hydrogen atom.
Erwin Schr¨odinger, 1926

Freeman Dyson writes in his foreword to Odifreddi’s book:
One of the most profound jokes of nature is the square root of −1 that the
physicist Erwin Schr¨odinger put into his wave equation in 1926 . . . The
Schr¨odinger equation describes correctly everything we know about the behavior
of atoms. It is the basis of all of chemistry and most of physics. And
that square root of −1 means that nature works with complex numbers.
This discovery came as a complete surprise, to Schr¨odinger as well as to
everybody else. According to Schr¨odinger, his fourteen-year-old girlfriend
Itha Junger said to him at the time: “Hey, you never even thought when
you began that so much sensible stuff would come out of it.” All through
the nineteenth century, mathematicians from Abel to Riemann and Weierstrass
had been creating a magnificent theory of functions of complex variables.
They had discovered that the theory of functions became far deeper
and more powerful if it was extended from real to complex numbers. But
they always thought of complex numbers as an artificial construction, invented
by human mathematicians as a useful and elegant abstraction from
real life. It never entered their heads that they had invented was in fact
the ground on which atoms move. They never imagined that nature had
got there first.

7.4.6 Heisenberg’s Uncertainty Principle
In 1927 Heisenberg discovered that there exists a deep difference between classical
mechanics and quantum mechanics. He derived the following fundamental result
in quantum physics:
The classical notion of the trajectory of a particle, which has a precise
position and a precise velocity at the same time, is not meaningful anymore
in quantum mechanics.
Explicitly, for the operators Q, P : S(R) → L2(R) called position operator Q and
momentum operator P, we have the Heisenberg commutation relation
(QP − PQ)ϕ = iϕ for all ϕ ∈ S(R). (7.53)
Let ϕ ∈ S(R) be a normalized state in the Hilbert space L2(R). We claim that
ΔxΔp ≥ h/2. (7.54)
This means that it is impossible to measure precisely the position and the momentum
of the quantum particle in the state ϕ at the same time. The uncertainty
inequality (7.54) follows from (7.53) as a special case of Theorem 10.4 on page 524
of Vol. I.

7.4.7 Unstable Quantum States and the Energy-Time
Uncertainty Relation
In particle accelerators, many particles are unstable; such so-called resonances
only live a very short time.
Folklore
We are going to show that wave packets are unstable in quantum mechanics. There
exists a fundamental inequality between the life-time of the wave packet and its
mean energy fluctuation which is called the energy–time uncertainty relation.

7.5 Feynman’s Quantum Mechanics
It is a curious historical fact that quantum mechanics began with two
quite different mathematical formulations: the differential equation of
Schr¨odinger, and the matrix algebra of Heisenberg. The two, apparently
dissimilar approaches, were proved to be mathematically equivalent. These
two points of view were destined to complement one another and to be ultimately
synthesized in Dirac’s transformation theory.
This paper will describe what is essentially a third formulation of nonrelativistic
quantum theory. This formulation was suggested by some of
Dirac’s remarks concerning the relation of classical action to quantum
mechanics. A probability amplitude is associated with an entire motion of
a particle as a function of time, rather than simply with a position of the
particle at a particular time.
The formulation is mathematically equivalent to the more usual formulations.
There are, therefore, no fundamentally new results. However, there
is a pleasure in recognizing old things from a new point of view. Also, there
are problems for which the new point of view offers a distinct advantage.40
Richard Feynman, 1948
The calculations that I did for Hans Bethe, using the Schr¨odinger equation,
took me several months of work and several hundred sheets of paper.
Dick Feynman (1918–1988) could get the same answer, calculating on a
blackboard, in half an hour.
Freeman Dyson, 1979

The idea is to
pass from time t to imaginary time it and to use analytic continuation in order to
translate well-known results from diffusion processes to quantum processes. This is
called the Euclidean strategy in quantum physics. The following golden rule holds:
Apply analytic continuation only to such quantities that you can measure
in physical experiments.
Analytic continuation of functions depending on energy plays a crucial role in studying
the following subjects:
• scattering processes,
• the energies energies of bound states, and
• the energies of unstable particles having finite lifetime (called resonances).

7.5.1 Main Ideas
The basic idea of Feynman’s approach to quantum mechanics is
• to describe the time-evolution of a quantum system by an integral formula, which
is equivalent to the Schr¨odinger differential equation,
• and to represent the kernel K(x, t; y, t0) of the integral formula by a path integral.
From the physical point of view, Feynman emphasized that
The description of quantum particles becomes easier if we use probability
amplitudes as basic quantities, but not transition probabilities.
The reason is that, in contrast to transition probabilities, probability amplitudes
satisfy a simple composition rule (also called product rule) which is at the heart
of Feynman’s approach to quantum theory. In terms of finite-dimensional Hilbert
spaces, the following hold:
• Feynman’s probability amplitudes are precisely the complex-valued Fourier coefficients
c1, c2, . . . , cn of a state vector.
• Feynman’s composition rule for probability amplitudes coincides with the Parseval
equation (7.81) for Fourier coefficients in mathematics.

According to Feynman, the passage from classical mechanics to quantum
mechanics corresponds to a statistics over all possible classical
paths where the statistical weight exp(iS[q]/h) depends on the classical action.
This is a highly intuitive interpretation of the quantization of classical processes.
Let us discuss the intuitive background.

7.6 Von Neumann’s Rigorous Approach
Rigorous propagator theory is based on von Neumann’s operator calculus
for functions of self-adjoint operators.
Folklore
As a preparation for the rigorous propagator theory to be considered in the next
section, let us summarize von Neumann’s operator calculus. In this section, we consider
an arbitrary complex separable Hilbert space X of finite or infinite dimension.
The inner product on X is denoted by ψ|ϕ for all ϕ, ψ ∈ X.

The Fourier–Laplace
transform is also briefly called the Laplace transform.
Interestingly enough, both retarded (i.e., causal) propagators and advanced
(i.e., non-causal) propagators play a crucial role in quantum field theory.
From the mathematical point of view, the reason is that the relevant perturbation
theory depends on quantities which are constructed by using both retarded and
advanced propagators. Physicists interpret this by saying that
• the interaction between elementary particles is governed by virtual particles
(which are graphically represented by the internal lines of the Feynman diagrams),
and
• the virtual particles violate basic laws of physics (e.g., the relation between energy
and momentum or causality).

7.6.4 The Free Quantum Particle as a Paradigm of Functional
Analysis
Extend the pre-Hamiltonian to a self-adjoint operator on an appropriate
Hilbert space X of quantum states, and use costates related to a Gelfand
triplet with respect to X.
The golden rule
The modern theory of differential and integral equations is based on functional
analysis, which was created by Hilbert (1862–1943) in the beginning of the 20th
century. The development of functional analysis was strongly influenced by the
questions arising in quantum mechanics and quantum field theory. In this section,
we want to study thoroughly how the motion of a free quantum particle on the real
line is related to fundamental notions in functional analysis.
This is Ariadne’s thread in functional analysis.
This way, the formal considerations from Sect. 7.5.3 will obtain a sound basis for
the free quantum particle.
The main idea of the modern strategy in mathematics and physics consists in
describing differential operators and integral operators by abstract operators related
to generalized integral kernels.
(i) In the language used by physicists, this concerns the Dirac calculus based on
Dirac’s delta function and Green’s functions (also called Feynman propagators).
(ii) In the language used by mathematicians this is closely related to:
• Lebesgue’s passage from the Riemann integral to the Lebesgue integral based
on measure theory in about 1900;
• von Neumann’s passage from formally self-adjoint operators to self-adjoint
operators and his generalization of the classical Fourier transform via spectral
theory in the late 1920s;
• Laurent Schwartz’s theory of generalized functions including the kernel theorem
in the 1940s;
• the generalization of von Neumann’s spectral theory by Gelfand and Kostyuchenko
in 1955 (based on quantum costates as generalized functions and
the corresponding Gelfand triplets);
• the extension of the Gelfand–Kostyuchenko approach to general nuclear
spaces by Maurin in 1959.
Tempered Distributions
In order to translate the very elegant, but formal Dirac calculus into mathematics,
one has to leave the Hilbert space of states used by von Neumann
in about 1930. Folklore
In what follows, we will use
• the space S(R) of smooth test functions ϕ : R → C which decrease rapidly at
infinity,
• and the space S
(R) of tempered distributions introduced on page 615 of Vol. I.
Our basic tools will be
• the Fourier transform and
• the language of tempered distributions, and Gelfand triplets.