Path Integrals and Quantum Processes

By Mark S. Swanson

This graduate-level text offers a systematic presentation of the path integral approach to calculating transition elements, partition functions, and source functionals. Topics include Grassmann variables, field and gauge field theory, perturbation theory, and nonperturbative results. Requires only some familiarity with quantum mechanics. Numerous exercises. Ideal as course supplement or for independent study. 1992 edition.

• Language: English
• Publisher: Dover Publications
• Release date: Feb 19, 2014
• ISBN: 9780486782300

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Chapter 1

Mathematical Preliminaries

Throughout this book there are numerous commonly used mathematical conventions and facts that are assumed to be familiar to the reader. This chapter reviews some aspects of these conventions and facts, but it is in no way intended as a comprehensive or systematic explication of these areas of mathematical physics. At first glance the topics covered may appear to be unrelated to each other; however, each will be essential to understanding the path integral. It is hoped that this chapter will clarify notation and refresh the reader’s memory regarding the facts presented. The reader who wishes further details should consult the references [1, 2].

The central theme of this book is the functional formulation of quantum processes. At the heart of this method is the concept of a complete set of functions. In order to define what is meant by completeness and functional differentiation it is first necessary to introduce the Dirac delta and its properties. This is done in Sec. 1.1. In Sec. 1.2 the concept of a complete set of functions is developed within the specific context of Fourier series and transforms. The representations of the Dirac delta can then be written in terms of a basis set of complete functions for the general class of functions under consideration. In Sec. 1.3 functionals are defined and a method of functional differentiation is formulated. These methods are then used to give a brief review of the action functional approach to classical mechanics. In Sec. 1.4 the most important properties of matrices are outlined and extended to functions of multiple variables. The chapter closes in Sec. 1.5 with a discussion of Gaussian integrals and Jacobians, since these are of paramount importance to the standard evaluation of the path integral.

1.1 The Dirac Delta

The Dirac delta δ(x) is defined by its integral [3]. Given a function f(x), the Dirac delta satisfies the integral relation

if the range of integration contains an open interval around the point x = α; if it does not, then the integral vanishes. The function f(x) is called a test function. The Dirac delta is a generalization of the concept of a function to a larger class of mappings, referred to as distributions [4]. For the sake of intuition the Dirac delta can be visualized as an improper function that is essentially zero everywhere except at the set of points where its argument vanishes, and at those points it is singular (infinite). This intuitive picture would lead immediately to the conclusion that the distribution xδ(x) = 0, but this fails when the test function against which it is integrated is singular at the origin, e.g., f(x) = 1/x. Clearly, if the set of test functions does not include functions singular at the origin, then the distribution xδ(x) does vanish when it is integrated. It is apparent from this simple observation that the properties of distributions can depend crucially upon the properties of the test functions against which they are integrated.

Deferring to the next section a more substantial definition of δ(x) and the test functions, numerous properties of the Dirac delta follow from the defining relation (1.1), and these are stated without detailed proof. Integration by parts, dropping the endpoint contributions, gives

A simple change of variables in the integral gives

If the argument of the Dirac delta is itself a function g(x), and the set of values x1, … , xn are the n roots of g(x), then it follows that

where g'(xi) is the first derivative of g at xi.

Exercise 1.1: Prove (1.4) by using (1.3) and a Taylor series representation of the function g(x) near its roots, assuming that all the roots of g(x) are simple, i.e., that the first derivative of g(x) is nonzero at each root.

The previous relations may be extended to multidimensional integrals, and δn(x) denotes the Dirac delta for an n-dimensional volume. In Cartesian coordinates δn(x) is the product of n one-dimensional Dirac deltas, each of which corresponds to one of the Cartesian coordinates. Such a product of Dirac deltas satisfies

where α is a point in the n-dimensional volume. The superscript on the Dirac delta will be suppressed whenever its dimension is apparent from the context of its use.

Exercise 1.2: Given the form of the volume element in spherical coordinates in n dimensions, dnx = rn–1 dr dø sin θ1 dθ1 … sinn–2 θn–2 dθn–2, express δn(x) in spherical coordinates as a product of one-dimensional Dirac deltas.

1.2 Completeness

It will be seen that a central feature of the path integral formulation of quantum processes is its use of functionals [5]. The general concept of a functional will be defined in detail in the next section. However, as its name implies, the functional is constructed from functions, and many times these functions are arbitrary members of a general class of functions. It is therefore extremely useful to have a method of expressing an arbitrary member of a general class of functions compactly. It is the property known as completeness that allows such a function to be constructed from a subset of well-understood functions by writing it as an infinite series, or an integral, involving those functions.

As an example, let f(x) be a member of the general group of real functions that are piecewise continuous and periodic on the interval x = (–L, L). It is known from the theory of Fourier series [6] that such a function may be written in the form

where the coefficients of the series are given by

The set of functions {sin(nπx / L), cos(nπx / L)} is said to be complete. In other words, it is a collection of functions sufficient to construct any member of the general class of functions that are periodic and piecewise continuous over the interval.

These formulas are exactly analogous to the representation of a general vector in a vector space in terms of its components along a set of basis unit vectors. That a general vector may be represented this way is possible because every vector space possesses an inner or dot product. This inner product allows a determination of the projection or overlap of two vectors onto each other, so that two vectors x and y are orthogonal if and only if x . y = 0. If {ei } is a set of orthonormal vectors with respect to the inner product, then ei . ej = δij. The symbol δij is the Kronecker delta, which is 1 when i = j and zero otherwise. If an arbitrary vector x in the space under consideration can be expressed as a sum of vectors along each basis vector in the form

where xi = x ei, then the set {ei} forms a complete orthonormal basis for the vector space. In the Cartesian case the {xi} are the coordinates of the tip of the vector.

The extension of these simple ideas to the Fourier series lies in generalizing the inner product to include functions. Defining an inner product for functions will allow them to be treated in a manner similar to vectors. To this end, the inner product of two functions, f(x) and g(x), that are periodic over the interval x = (– L, L) is defined as

where f* is the complex conjugate of f. Such a definition satisfies the general properties of an inner product: f.f > 0 (positive-definite norm), (f + g).(f + g) ≥ f.f + g.g (triangle inequality), and |f-g|² ≤ (f.f)(g.g) (Cauchy–Schwarz inequality). Using this definition, it is easy to show that the sine and cosine functions used in the Fourier series (1.6) form an infinite dimensional orthonormal set of functions indexed by the integer n. This follows from the fact that the integrals of these functions satisfy

for m, n > 0. Thus, the result of taking the inner product of any two of these functions is identical to that of the inner product of the basis vectors for a vector space. The quantities an + a–n and bn – b–n, obtained from (1.7) and (1.8), are the projections of the function f(x) onto these basis functions. The Fourier series is then seen as the generalization of the component representation of a vector, familiar from finite dimensional vector spaces, to an infinite dimensional function space. For the space of periodic functions, the sine and cosine functions play the role of orthonormal basis vectors and span the entire function space, while the combinations an + a–n and bn – b–n can be visualized as the coordinates of the function in the function space. With a few additional subtleties that will be ignored here, both the simple finite dimensional vector space discussed in the previous paragraph and the infinite dimensional function space of this paragraph are examples of a Hubert space [7].

The Dirac delta of Sec. 1.1 may now be given a representation in terms of a Fourier series. If x, y ∈ (–L, L), then (1.6), (1.7), and (1.8) give

It is clear that this representation of the Dirac delta is itself periodic, and therefore its test function space is limited to the set of functions periodic over the same interval. If it is integrated over the interval (–L, L) against a function with a different periodicity, then it is not necessarily true that (1.1) will be satisfied.

Exercise 1.3: Using the periodic form (1.14) for the Dirac delta, show that

It is worth noting that different subclasses of functions may require a different set of basis functions. For example, the set of functions that are reflection symmetric, i.e., f(–x) = f(x), do not require any sine terms in their expansion. As a result, if the test function space is restricted to this subclass, then the representation of the Dirac delta need only contain the cosine terms.

Very often a function has no periodicity. Another way of expressing this is to say that the interval of periodicity is infinite, and this is represented mathematically by taking the limit L → ∞ in the previous formulas. In this limit the sum in (1.14) may be converted into an integral with the result that

Exercise 1.4: Derive expression (1.16).

Using property (1.1) for a function f(x) immediately gives

(k) is called the Fourier transform of f(x), and is given by

Result (1.17) shows that eikx defines a continuous set of functions, and that this set is complete for the space of nonperiodic functions. It is more common in the literature to see (1.14) and (1.16) used as an assertion that the respective sets of functions are complete. The relation (1.1) is then used to define the Fourier series, or transform, of an arbitrary function by integrating the appropriate form of the Dirac delta against it. However, the two versions of the Dirac delta are quite different, and so it is more correct to make the argument in reverse, as was done here. In this respect it is important to note that eikx is a complex valued function, and that the test function space now includes complex functions. From this point forward the limits on the Fourier transform integrals will not be explicitly displayed, unless they differ from infinity.

Exercise 1.5: Let f(x) be a member of the class of functions that are nonsingular in an open interval around the origin x = 0. Show that, for such a test function,

so that the limit of the exponential function behaves as δ(x).

The Fourier series and transform are generalized to many dimensions by simply adding additional degrees of freedom. The Dirac delta appropriate for n dimensions and nonperiodic functions is given by

where x, y, and k are vectors in the respective spaces and the integration is over the entire volume of k space. This form of the Dirac delta can be used to define the Fourier transform of a chosen function by simply integrating it against that function.

There may be many complete sets of functions that span the same function space, just as there are many choices for a basis set of unit vectors in a vector space. The choice of a set is often a matter of either calculational convenience or selection of a basis that best utilizes an underlying symmetry in the system being analyzed. However, one of the most important aspects of these complete sets of functions is that many, if not all, originate from solving the eigenvalue problem for some Hermitian differential operator. This result can be sketched by using D(x) to denote a differential operator in one dimension, i.e., some combination of x and derivatives that has the property that, for any two suitable functions,

where D* is the complex conjugate of D. If D satisfies (1.20), then D is said to be Hermitian. The functions f and g for which (1.20) is true define two function spaces. If these two function spaces coincide, then D is said to be self-adjoint. For example, i∂/∂x is Hermitian, while ∂/∂x is not. For many physically relevant Hermitian differential operators, in particular the second-order Hermitian linear differential operators, it is known that the solutions of the associated eigenvalue problem,

subject to appropriate boundary conditions on the eigenfunctions φn or eigenvalues λ(n), are orthonormal,

and form a complete set,

In general, the proof of completeness reduces to showing that a Fourierlike series in the eigenfunctions converges to an arbitrary function in the general class. The proof will not be given here.

Exercise 1.6: Prove that the eigenvalues of a Hermitian operator are all real, and that the eigenfunctions associated with different eigenvalues are orthogonal.

For example, the Hermitian differential operator D = ∂²/∂x², along with the boundary condition that the eigenfunction φn is periodic, gives the familiar sine and cosine functions of the Fourier series, as well as the associated eigenvalues λn = –(nπ/L)². As another example, the differential operator D = –i∂/∂x, along with the demand that the eigenvalues be real, gives the complete set of eigenfunctions eikx with the eigenvalue k. Thus, the pairing of Hermiticity and real eigenvalues leads immediately to a complete set of eigenfunctions.

As a final remark, it often happens that the set of eigenfunctions contains both a discrete and a continuous set of eigenfunctions. Such a result may be expected on purely physical grounds. For example, in quantum mechanics the set of discrete eigenfunctions of the differential operator representing the energy corresponds to bound or negative energy states, while the continuous spectrum of eigenfunctions describes the scattering or positive energy solutions. It is reasonable to expect to find quantum mechanical systems whose solutions give both discrete and continuous eigenfunctions, since there are many physically meaningful potentials that can both bind and scatter particles. If {φk(x)} is the continuous set and {φn(x)} the discrete set, then the general statement of completeness in such a case in one dimension is given by

1.3 Functionals

A real-valued function is a mapping from some space into the real numbers. According to the last section it is possible to view a function itself as a point in an infinite dimensional space. Using this analogy, a functional is defined as a map of a function or a polynomial of functions into a number. The function plays the role of the coordinates being mapped by the functional. The standard method of constructing a functional is by integrating a collection of functions and their products over some interval of their arguments.

Determining the nature of the functions that extremize the functional historically led to the calculus of variations [8]. By analogy to the problem of finding the extrema of a function, the determination of the form for the extremum functions requires the definition of functional differentials and differentiation. Once these definitions have been made, it is then possible to create a local or pointwise criterion that extremum functions must satisfy. In order to make this definition the simplest case is considered initially, so that the general functional of a single function g(x[g(xwith respect to g(y) is then formally denned as [9, 10]

where δ(x – y) is the Dirac delta of appropriate dimension to match the dimension of integration.

Exercise 1.7: Using ([g(xdx[g(x)]n, has the functional derivative

if y ∈ (a, b), and is zero if y ∉ (a, b).

The definition (are two functionals, then their product has the functional derivative

which is the Leibniz property.

[g] is a functional that is well-behaved in the interval in function space around g has a Taylor series representation given by

where the limits of integration will be suppressed from now on. Applying an arbitrary power of the functional derivative to both sides of (1.28) and evaluating the result at g = 0 shows that (1.28) is self-consistent. Functional Taylor series may be defined around other functions, so that

Form (1.29) allows a functional to be expanded about a function for which the functional derivatives are well-defined.

Form (1.29) also allows the definition of a functional differential [11]. If δg(x) is an infinitesimal deviation from the function g(x), so that the function g(x) + δg(x) is infinitesimally close to g(x) everywhere, then the functional Taylor series (1.29) gives

where terms of order (δgunder an infinitesimal variation of its argument may then be defined as

The definition (1.31) is exactly analogous to the standard differential for a function of many variables. There the differential of a function f(x) is defined as the change of the function under an arbitrary change in the coordinates dx, where, in the case of many variables, dx is understood to be an infinitesimal vector. The differential of f(x) is then written

The result (1.31) is seen to be identical to (1.32) when the form of the function space inner product (1.10) is recalled. Thus, δg(x) plays the role of dx, and the functional derivative is effectively the gradient of the functional.

[g], where g is in turn a functional of f, so that g = g[f]. Two consecutive applications of the differential operation yields

By comparison with the standard form (1.31) for the differential, it follows that

has for an argument a set of functions {gi(x)}, and these are indexed by a subscript for ease of notation. At this point it is convenient to introduce the summation convention. In this convention any repeated pair of indices in a formula is understood to stand for a summation over the entire range of the indices, unless one of the indices is enclosed in parentheses. As an example, the inner product of two n-dimensional real vectors, x and y, can be written

in terms of their Cartesian coordinates. However, the expression λ(i)xi does not represent a summation, rather simply the product of λ(i) with xi Where there is any chance of confusion the summation will be explicitly displayed. Using the summation convention, it follows that the differential of the functional of many functions may be written

where the sum over j runs from 1 to n.

The Lagrangian formulation of Newtonian mechanics is one of the earliest applications of functional methods in physics [12], and it serves as an excellent example. In the simplest nontrivial case a point particle of mass m is constrained to move one dimensionally in the presence of a conservative force F(x). Newton’s law of motion states that

where V(x) is the potential energy associated with the force, and ẍ is the second derivative of x with respect to the time t. Combining the solution of (1.37) with appropriate initial conditions gives a unique function x = x(t), which represents the particle’s trajectory. The solution to Newton’s equation of motion for a given set of initial conditions will be referred to as a classical trajectory.

In the Lagrangian formulation of mechanics the object of importance is not the differential equation of (1.37), but rather the action functional, which is formed from the kinetic energy and the potential energy of the particle’s motion. This is simply the integral of the Lagrangian density, which is defined to be the difference between the kinetic energy and the potential energy. In the simple example defined by (1.37) the action functional is given by

where ta and tb define the time interval for the motion of the particle. At this point x(t) is considered to be an arbitrary function of the time. Once a well-behaved form for x(t) is chosen the action functional can be evaluated to give a value.

Newton’s equation of motion is recaptured by asking for the criterion that the function x(t) must satisfy in order to make the action an extremum. Exploiting the analogy between functions and functionals shows that this is equivalent to demanding that the differential of the action vanish around the extremizing form for x(t). Using the result that

and the functional chain rule (1.34), it follows that the extremizing function must be chosen such that

where δx(t) is an arbitrary infinitesimal variation which vanishes at ta and tb. Using the results of Exercise 1.7, it follows that the form for x(t) that extremizes the action must obey the Euler‒Lagrange equation,

which is the generalization of Newton’s law of motion. The definitions of the functional derivative and the functional differential allow a very compact statement of the calculus of variations, which was originally developed to solve the same problem of extremizing functionals.

If the only use for the action functional approach to mechanics was to generate the already known equations of motion, then it would be considered nothing more than an oddity. However, the action approach to formulating the dynamics of a physical system contains far more information than simply the equation of motion. First, the action functional is a global statement about the system, from which a local differential equation can be derived by demanding an extremum. The action is a global object in the sense that it receives contributions from the entire trajectory of the particle, and hence it records an aspect of the history of the particle’s motion. Second, although the action is extremized by the set of classical trajectories that solve (1.41), the action can be evaluated for any trajectory. It will be seen that this is of profound importance in quantum processes. Third, the action approach allows the definition of the momenta canonically conjugate to the coordinates of the particle, and the generalization of the energy known as the Hamiltonian can be constructed from these by means of a Legendre transformation. Fourth, the action contains all the information regarding the classical symmetries of the system, and these determine the conservation laws associated with the classical trajectory. As a result, the action approach is the most compact and systematic method to formulate the dynamics of a physical system.

In order to construct a general form for the Hamiltonian, the Lagrangian density will be written as a function of a set of n generalized coordinates, q = {qii). The momentum pi, canonically conjugate to the coordinate qi, is defined as

It is assumed that (1.42) can be solved for all the , . in terms of the set of 2n variables {pi, qii i(p, q). The Hamiltonian H(p, q) is then defined by a Legendre transformation as

Using (1.41) and (1.42) shows that Η has no explicit time dependence since it satisfies

when it is evaluated along a classical trajectory. The Hamiltonian is then seen as the generalization of the energy for the classical system’s motion.

The definition (1.43) may be reversed to give another variant of the action functional, expressed in terms of the 2n degrees of freedom p and q. In this version the action is written

Again demanding that δS = 0 under simultaneous variations of p and q gives

Since the variations δpi and δqi are independent, the original n second-order differential equations generated by the Euler-Lagrange formula (1.41) have become the 2n first-order differential equations

These are known as Hamilton’s equations of motion.

Both form (1.38) and (1.45) represent the same dynamical system. However, there is a subtle difference between the two, and this difference can be be important. In constrained systems form (1.45) is more useful, particularly for momentum (velocity) dependent constraints. Because the form (1.45) has both p and q present, it can serve as the starting form of the action when a canonical transformation of variables is performed. A canonical transformation can be defined as any transformation on the p and q variables that has a Jacobian, to be discussed in Sec. 1.5, equal to unity.

An equivalent definition of a canonical transformation is any transformation of p and q that leaves the Poisson bracket invariant. In classical mechanics the Poisson bracket of any two functions A(p, q) and B(p, q) is the antisymmetric operation defined by

Using the definition of the Poisson bracket and Hamilton’s equations of motion, it is straightforward to show that

and

It is the Poisson bracket formalism that carries over into quantum mechanics to become the commutator (see Sec. 2.1). The Poisson bracket has the important property that it reproduces the quantum mechanical behavior of p and q from their classical forms. This will render it of great value in the functional approach.

1.4 Matrices

This section contains a very brief review of the properties of matrices and their eigenvalues [13]. A general matrix Μ is an n × m array of complex numbers called the elements of M. For the purposes of this book consideration will be limited to square n × n arrays, Μ, where n is the rank of the matrix, and single column n × 1 arrays, x, where n is the dimension. The latter will be referred to as vectors because of their analogy with the component form of the vectors familiar from configuration space. The individual elements of a matrix will be denoted Mij, and those of a vector xij, so that i is the row and j the column of the element. A matrix is said to be real if every element is a real number. Two matrices are equal only if each of their respective elements are equal. The elements of the sum of two matrices M and N are given by the sum of the elements

while the elements of the product of the two matrices are given by (summation convention)

The commutator of two matrices is defined as

Two matrices, M and N, are said to commute if [M, N] = 0.

Such a definition again guarantees that any nonzero vector will have a positive norm. Two vectors satisfying x . y = 0 are said to be orthonormal.

αβγ… which has n superscripts that range over the values {1, … , n}. It is completely antisymmetric so that it is defined to be +1 if the set {α, β, …} is an even permutation of the first n integers and —1 if it is an odd permutation. Since it is completely antisymmetric, the Levi–Civita symbol vanishes if any two superscripts coincide in value. The determinant of an n × n matrix M is defined as

Any matrix with a determinant of unity is said to be unimodular.

Exercise 1.8: Using definition (1.54) show that

and that det (MN) = det M det N

Another important number associated with a matrix is the trace. The trace of a matrix M is denoted Tr M, and is given by

The trace of the product of a set of matrices has the property that it is invariant under cyclic permutations of the product.

Exercise 1.9: Show that Tr (M1 MM2 … M n) = Tr (Mn M1 … Mn–1).

The identity matrix, denoted I, has the elements Iij = δij. It is trivial to show that IM = MI = M, so that I commutes with all other matrices. A matrix M has a left inverse M = I. A right inverse. Clearly, det I = 1 and Tr (I) = n. From Exercise 1.8 it then follows that det MM–1 = det M det M–1 = 1, so that M–1 exists only if det M ≠ 0.

In complete analogy to differential operators acting on functions the square matrices can be viewed as operators acting on the set of vectors. If M is an n × n matrix, then an eigenvector x satisfies the equation

where, in formal analogy with (1.21), λ is called an eigenvalue of M. It can be shown that each eigenvalue must satisfy the equation

Since (1.58) generates an nth order polynomial, there can be at most n distinct eigenvalues associated with an n × n matrix. Once the eigenvalues of a matrix are known, it is a straightforward algebraic problem to find the eigenvectors by solving (1.57). The eigenvectors so determined will be unique only up to a multiplicative constant, but this constant may be fixed, up to an overall complex factor of modulus one hereafter referred to as a phase= 1.

Given an arbitrary n × n matrix M, two new matrices may be defined by interchanging rows and columns. The transpose of the matrix M is denoted MT, and its elements are given by (MT)ij = Mij. A vector, or column matrix, becomes a row matrix under the transpose operation. The Hermitian adjoint . Under the Hermitian conjugation operation a vector, or column matrix, becomes a row matrix with elements given by the complex conjugate of the original vector elements. It is easy to show that, if M and N are two equal rank square matrices, then (MN)T = NTMTand (MN)† = N†M†.

Using these definitions matrices may be classified in the following ways. A matrix O is said to be orthogonal if it obeys OOT = I, so that OT = O–1. A matrix S is said to be symmetric if it satisfies S = ST. A matrix A is said to be antisymmetric if AT = –A. A matrix U is said to be unitary if it obeys UU† = I, so that U† = U–1. A matrix H is said to be Hermitian if it satisfies H = H†.

Given a unitary matrix U, another n × n matrix M may be used to create the matrix M' by a unitary transformation, defined by

If the matrix M is unitary, orthogonal, or Hermitian, then the unitarily transformed matrix M' will retain these properties. Using the previous results shows that det M' = det M and Tr M = Tr M'.

The set of n normalized eigenvectors that solve (1.57) are denoted {x(j)}, and each x(j) has n components, denoted xi(j).

Exercise 1.10: Show that, if M is a Hermitian matrix, then its eigenvalues are real and its eigenvectors belonging to different eigenvalues are orthonormal.

Exercise 1.11: Show that the eigenvalues of a real antisymmetric matrix are pure imaginary numbers, and that the determinant of a real n × n antisymmetric matrix vanishes if n is an odd number.

The results of Exercise 1.10 show that the matrix U, formed from the elements of the eigenvectors of the Hermitian matrix M and given by Uij = xi(j) is unitary. It follows that the elements of the unitary transformation of M by U are given by

where λ(j) is the jth eigenvalue of M. The unitarily transformed matrix has nonzero elements only along its diagonal; all other elements vanish. From relation (1.60), and the invariance of the determinant and the trace under a unitary transformation, it follows that

Thus, the necessary condition for the existence of the inverse of a matrix is simply the absence of any zero eigenvalues.

It is possible to define functions of matrices by using the Taylor series representation of the function. For example, a matrix may be exponentiated by using the power series

where Mn represents a product of n factors of M. It should be obvious from this example that the elements of the function of the matrix, derived by inserting the original matrix into the power series, will coincide with the function of the elements of the original matrix only if the original matrix is diagonal. In the event that the power series is not well defined this definition of the function of a matrix allows an expansion to be made about another matrix in order to ensure that the power series is sensible. For example, the logarithm function is not defined for the argument zero.

For this reason the power series representation of In M must be defined by expanding it around another nonzero matrix, typically the identity matrix I. The In M function can then be written

Exercise 1.12: Show that a Hermitian matrix M satisfies the relation

A useful result is the Baker-Campbell-Hausdorff theorem [14]. If C denotes the commutator of the two matrices A and B, so that C = [A, B], and if C commutes with both A and B, so that [C, A] = [C, B] =0, then the theorem states

This theorem exploits the fact that exp(A) exp(–A) = I. The first step in proving the theorem is to use the result that, for the conditions of the theorem,

Using the Taylor series representation of the exponential function along with result (167) shows that, for α a real variable,

Then the matrix, defined by

satisfies the equation

This differential equation has the solution

so that, at α = 1, the solution reduces to

thus demonstrating the Baker-Campbell-Hausdorff theorem.

It is very useful to complete the analogy between functions, vectors, and matrices. In this sense it is possible to view a function of a single variable, defined over the interval (–L, L), as a vector. In order to demonstrate this the inner product of two functions, defined by (1.10), is written as a Riemann sum by choosing a discrete set of points over the interval (–L, L). This gives

where xj = –L + j(2L/N). Relation (1.73) formally resembles the inner product of two infinite dimensional matrix vectors, defined in terms of the vector components and weighted by the measure factor 2L/N.

This analogy to vectors and matrices carries over into functions of two variables. If f(x, y) and g(x, y) are two arbitrary functions, then it is possible to define a third function h(x, z) by writing

By again making the interval (–L, L) discrete and writing (1.74) as a Riemann sum,

a form resembling the multiplication of two matrices is obtained. In this entirely nonrigorous manner the results of matrix analysis may be taken over into the realm of functions and functionals. The function f(x, y) can be viewed as the x, y element of a continuous matrix f.

In such a scheme the identity matrix can be represented functionally by a Dirac delta, since

Through this analogy with matrices the trace of the power of a function of two variables is defined as

A caveat is in order, since the action of integrating the function changes the units associated with the result by a factor of the units associated with the measure of the integral. In the cases presented so far the units of the measure dx appearing in the integrals have been length. The units of the Dirac delta are (length)–1, so that any power series representation of a function and Dirac deltas indicates that the function itself should also have units of (length)–1. In the