The Path Integral Formulation of Quantum Mechanics
We turn now to a topic that is important already for ordinary quantum mechanics and essential in quantum field theory: the so-called path integral. In the setting of ordinary quantum mechanics (of the sort we have been considering in this book), the integ
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We turn now to a topic that is important already for ordinary quantum mechanics and essential in quantum field theory: the so-called path integral. In the setting of ordinary quantum mechanics (of the sort we have been considering in this book), the integrals in question are over spaces of “paths,” that is, maps of some interval [a, b] into Rn . In the setting of quantum field theory, the integrals are integrals over spaces of “fields,” that is, maps of some region inside Rd into Rn . Formal integrals of this sort abound in the physics literature, and it is typically difficult to make rigorous mathematical sense of them—although much effort has been expended in the attempt! In this chapter, we will develop a rigorous integral over spaces of paths by using the Wiener measure, resulting in the Feynman–Kac formula. We begin with the Trotter product formula, which will be our main tool in deriving the path integral formulas. From there we turn to the (heuristic) path integral formula of Feynman, and then to the rigorous version of Feynman’s result obtained by M. Kac, the so-called Feynman–Kac formula. Although it is not feasible to give complete proofs of all results presented here, we give enough proofs to get a flavor of the mathematics involved. We will prove a version of the Trotter product formula and, assuming the existence of the Wiener measure, a version of the Feynman–Kac formula.
B.C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, DOI 10.1007/978-1-4614-7116-5 20, © Springer Science+Business Media New York 2013
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20. The Path Integral Formulation of Quantum Mechanics
20.1 Trotter Product Formula The Lie product formula (Point 7 of Theorem 16.15) says that for all X and Y in Mn (C), we have eX+Y = lim (eX/m eY /m )m . m→∞
The Trotter product formula asserts that a similar result holds for certain classes of unbounded operators on Hilbert spaces. Theorem 20.1 (Trotter Product Formula) Suppose that A and B are self-adjoint operators on H and that A+B is densely defined and essentially self-adjoint on Dom(A) ∩ Dom(B). Then the following results hold. 1. For all ψ ∈ H, we have lim eit(A+B) ψ − (eitA/N eitB/N )N ψ .
(20.1)
2. If A and B are bounded below, then for all ψ ∈ H, we have lim e−t(A+B) ψ − (e−tA/N e−tB/N )N ψ .
(20.2)
N →∞
N →∞
In both results, the expression A + B refers to the unique self-adjoint extension of the operator defined on Dom(A) ∩ Dom(B). In the usual terminology of functional analysis, (20.1) asserts that the operators (eitA/N eitB/N )N converge to eit(A+B) in the “strong operator topology,” and similarly with (20.2). We will give a proof of this result in the special case in which A + B is densely defined and self-adjoint on Dom(A) ∩ Dom(B). This condition holds, for example, whenever the Kato–Rellich theorem (Theorem 9.37) applies. See Sect. A.5 of [14] for a proof of the version stated above. Proof. Since all the operators in Point 1 of the theorem are unitary, it is easy to see that if the result holds on some dense subspace W
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