Determinants of Quantum Operators Nicholas Wheeler, Reed College Physics Department November 2009 Introduction. Justin Sumner has elected to attempt to develop a thesis from material that he encountered in a recent issue of AJP.1 The paper concerns the evaluation of the “functional determinants” that appear as “preexponential” factors when Feynman’s “sum-over-paths” formalism is used to construct quantum mechanical propagators, and that occur also in some other closely related contexts. The authors may be technically correct when they assert that “the prerequisites for this analysis involve only introductory courses in ordinary differential equations and complex variables,” but neglect to mention that the work is certain to seem entirely unmotivated to readers whose (quantum mechanical) experience happens not to have exposed them to certain fairly arcane issues/problems. Here my primary objective will be to motivate the subject, to place it in an intelligible physical/mathematical context. Initially I will be content to wander about on the periphery of the subject, reminding myself of issues/methods that seem to be of some remote relevance. Free particle propagator.2 The equation ∂ ψ = aD2ψ : D ∂ (1) ∂t ≡ ∂x becomes (i) the free particle Schroding¨ er equation if we set a = i ! 2m 1 Klaus Kirsten & Paul Loya, “Calculation of determinants using contour integrals,” AJP 76, 60–64 (2008). 2 Here I borrow from the opening paragraphs of my research notes: Theory and physical applications of the Appell transform (). That work was inspired by material I encountered in D. V. Widder’s The Heat Equation (). 2 Determinants of quantum operators and interpret ψ(x, t) to be a probability amplitude; (ii) the diffusion equation if we set a = diffusion coefficient and consider ψ to to provide a description of the relevant density; (iii) the Fourier heat equation if we set thermal conductivity a = (specific heat)(density) Formally, we expect—whatever the interpretation—to have at D 2 ψt(x) = e ψ0(x) (2) Familiarly, one has the Gaussian integral formula + ∞ (αx2+βx) β2/4α e− dx = π/α e : (α) > 0 (3.1) · " !−∞ " of which (send α 1/4a) #−→ + aβ2 1 ∞ 1 a–1x2 βx e = e− 4 − dx (3.2) √4πa !−∞ is a notational variant—of present interest because the expression on the right can be considered to provide an integral representation of the expression on the left. We proceed on the tentative assumption that it makes formal sense to write + at D 2 1 ∞ 1 ξ2 ξ D 1 e = e− 4at − dξ : ( > 0) (4) √4πat " 4at !−∞ —the point here being that the operator D, which appears squared on the left, appears unsquared on the right. By Taylor’s theorem we have (within some unstated radius of convergence) ξ D e− f(x) = f(x ξ) − so we expect to have + 1 ∞ 1 ξ2 ψt(x) = e− 4at ψ0(x ξ) dξ √4πat − !−∞ + ∞ = g (ξ) ψ (x ξ) dξ t 0 − !−∞ 1 1 x2 where gt(x) e− 4at (5) ≡ √4πat Free particle propagator 3 A final change of variables ξ y = x ξ gives #−→ − + ∞ ψ (x) = g (x y) ψ (y) dy (6) t t − 0 !−∞ The function ψt(x) thus produced satisfies (1) because each member of the y-parameterized family of functions g (x y) does: t − ∂ g (x y) = aD2g (x y) (7.1) ∂t t − t − And lim ψt(x) = g0(x) because lim gt(x y) = δ(x y) = δ(y x) (7.2) t 0 t 0 − − − ↓ ↓ The functions gt(x y) are the so-called “fundamental solutions” of the linear partial differential equation− (1). We might write gt t (x x0) = G(x, t; x0, t0) − 0 − to cast (6) into the form + ∞ ψt(x) = G(x, t; x0, t0) ψ0(x0) dx0 (8) !−∞ to emphasize that the fundamental solution is precisely the Green’s function of the differential equation in question. The prefactor 1 √4πat provides a trivial instance of the kind of object that the theory of functional determinants has been designed to evaluate. Higher-dimensional generalization. Let D2 be interpreted now to mean ∂1 ∂ 2 2 ∂ D = A where . with ∂i ∇· ∇ ∇ ≡ . ≡ ∂xi ∂n and where A is a real symmetric n n matrix (additional properties yet to be × specified). In the case A = I the second-order differential operator D2 reduces to the familiar Laplacian: 2 . Now, in straightforward generalization ∇·∇ of (3)—i.e., of ∇ ≡ ∇·∇ + 2 ∞ α x βx π 1 –1 e− ± dx = exp β α β α 4 !−∞ ) * + 4 Determinants of quantum operators —one has3 + n ∞ xiA xj b xi n π 1 b Aij b e− ij ± i d x = e 4 i j · · · ,det Aij !! !−∞ ) ) ij –1 where A Aij and summation on repeated indices is understood, and to assure) con) ≡vergence) ) we have had to assume that all the (necessarily real) eigenvalues of A are positive. More neatly, + ∞ πn 1 –1 x Ax b x n 4 b A b e− · ± · d x = e · (9) · · · det A !! !−∞ ) or again (send A 1 B –1) #−→ 4 + 1 –1 b B b 1 ∞ x B x b x n e · = e− 4 · ± · d x (4π)n det · · · B !! !−∞ " on which basis we expect to have (at least formally) + 1 –1 t A 1 ∞ ξ A ξ ξ n e ∇· ∇ = e− 4t · − ·∇ d ξ (10) (4π t)n det · · · A !! !−∞ " from which we recover (4) in the one-dimensional case. The multi-variable Taylor theorem provides (again within some unstated radius of convergence) ξ e− f(x) = f(x ξ) ·∇ − so we expect the solution of ∂tψ(x, t) = ( A )ψ(x, t) : ψ(x, 0) = ψ0(x) ∇· ∇ to be expressible t A ψ(x, t) = e ∇· ∇ψ(x, 0) + ∞ = g (ξ)ψ (x ξ) dξ dξ (12) · · · t 0 − 1 · · · n !! !−∞ 1 1 x A–1x where now gt(x) e− 4t · (13) ≡ (4π t)n det A " 3 See page 32 in my quantum mechanics (/), or (for example) http://en.wikipedia.org/wiki/Gaussian integral Problems that arise when n is very large 5 If A were diagonal a 0 0 1 · · · 0 a2 0 A = . ·.· · . . .. 0 0 an · · · —which can in principle always be achieved by a rotational change of variables— then (13) assumes the especially transparent form n –1 g (x) = 1 e xi(4ait) xi t √4πa t − i=1 i - In such cases, the n-dimensional problem resolves into what are essentially n copies of the one-dimensional problem.4 Problems that arise when n is very large. When n is very large (and especially in the limit n ) and spectral information is absent, the use of (13) presents two closely related→ ∞ problems: 1) the definition of det A (which involves a sum over signed permutations) becomes unworkable, and 2) so also (for similar reason) does the matrix-theoretic construction of A. Looking first to the former problem, we if we were in position to write A = exp L then we would have det A = exp tr L (14) which, if one commands (say) L=RandomReal[. / 0,1 , 5,5 ] to construct a { } { } random 5 5 with random elements drawn from{ the}in{terval} 0, 1 , is readily verified by×calculation. The identity (14) presents this advantage:{ ev}aluation of tr L involves no permutational combinatorics—is straightforward whatever the dimension of L. If spectral information were available, one would have det A = ai i - whence log det A = log ai i 0 4 My “essentially” becomes “literally” when and only when the initial value of ψ(x, t) factors: n ψ(x, t) = fi(xi) i=1 - 6 Determinants of quantum operators Now the non-obvious step which Kirsten & Loya1 take in the introduction to their paper, and which appears to be entirely characteristic of work in this s s log a field: trivially a− = e− , so d s s log a a− = e− log a ds − ↓ = log a at s = 0 − and we have log det A = log ai − − i * 0 + d s = ds ai− − s=0 i 1 0 1 Enlarging upon the definition of the Riemann ζ-function1 ∞ s ζ (s) n− Riemann ≡ n=1 0 we define ∞ s ζ (s) a− (15) A ≡ i i=1 0 which gives back ζRiemann(s) in the special case ai = i: (i = 1, 2, 3, . .). In this notation d log det A = ds ζ A(s) − s=0 1 = ζ $ (0) 1 − A 1 , det A = exp ζ $ (0) (16) − A Comparison with (14) supplies ζ (0) =.tr log . /These equations are pretty, A$ A but become useful only if one is−in position to speak without knowledge of the A-spectrum about relevant properties of the function ζ A(s). The preceding remarks have nothing per se to do with quantum mechanics, but do have quantum mechanical applications, and it is in the latter context that—for the reason just stated—one acquires interest in the general theory of quantum ζ-functions.5 5 Richard Crandall, in “On the quantum zeta function,” J. Phys. A: Math. Gen. 29, 6795–6816 (1996), draws attention to the remarkable fact that s Z(s) = Ei− i 0 can sometimes be evaluated exactly even though not a single eigenvalue Ei is known. He himself concentrates on cases in which s assumes (not arbitrary complex, but) integral values. The quantum connection 7 Basic notions. The dynamical evolution of an arbitrary quantum state ψ) is accomplished by action | ψ) ψ) = U(t , t ) ψ) | t0 #−→ | t1 1 0 | t0 of a unitary operator of which these are fundamental properties: U(t0, t0) = I (17.1) U(t2, t0) = U(t2, t1)U(t1, t0) (17.2) t Differentiation of the unitarity condition U(t, t0)U (t, t0) = I establishes that necessarily U(t, t0) satisfies a differential equation of the form ∂ i! ∂t U = HU : H hermitian, with [H] = energy (18) of which the Schr¨odinger equation ∂ i! ψ)t = H ψ)t ∂t | | is an immediate corollary.