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Introduction to Algebraic and Constructive Quantum Field Theory
By John C. Baez, Irving E. Segal, Zhengfang Zhou PRINCETON UNIVERSITY PRESS
Copyright © 1992 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08546-3
CHAPTER 1
The Free Boson Field
1.1. Introduction
Much of the quantum field theory is of a very general character independent of the nature of space-time. Indeed, a universal formalism applies whether or not there exists an underlying "space" in the usual geometrical sense. In its primary form, this universal part of quantum field theory depends only on a given underlying (complex) Hilbert space, say H. Colloquially, H is often called the single-particle space.
Thus, for a nonrelativistic particle in three-dimensional euclidean space R3, H is the space L2(R3) consisting of all square-integrable complex-valued functions on R3, in the usual formalism of elementary quantum mechanics. For a relativistic field or particle as usually treated, H is the space of "normalizable" wave functions. Here the norm derives from a Lorentz-invariant inner product in the solution manifold of the corresponding wave equation. For possible more exotic types of fields, the situation is much the same.
This chapter presents the mathematical theory of one of the most fundamental quantum field constructs from a given complex Hilbert space H, without at all concerning itself with the origin of H. This theory has close relations to integration and Fourier analysis in Hilbert space, and can in part be interpreted as the extension of analysis in euclidean n-space to the case in which n is allowed to become infinite. We call this universal construct the free boson (more properly, Bose-Einstein) field overH.
The next section of the chapter begins the rigorous mathematical development. From time to time, items logically outside the mathematical development, labeled Lexicon, will interrupt in order to correlate the somewhat abstract treatment with physical usage and intuition. Mathematical Examples, in the nature of special cases, will also be provided. Readers interested primarily in the mathematics may largely ignore the Lexicon items. Those who would like to appreciate at this point how the conventional treatment of relativistic fields can be subsumed under the universal Hilbert space formulation of this chapter will find an explicit treatment in Appendix B.
1.2. Weyl and Heisenberg systems
In any Hilbert space, the imaginary part of the inner product provides a real antisymmetric (real-) bilinear form. A more general type of space, in which only such a form is given, also plays an important part in boson theory.
Definition. A symplectic vector space is a pair (L, A) consisting of a real topological vector space L, together with a continuous antisymmetric, "nondegenerate" bilinear form A on L. To be more explicit, "nondegenerate" means that if A(x, y) = 0 for all x [member of] L, then y = 0.
When L is a real finite-dimensional vector space, it is easily seen that it can be given the structure of real topological vector space in one and only one way. A real infinite-dimensional vector space is a topological vector space relative to the topology in which a set is open if and only if its intersection with each and every finite-dimensional subspace is open relative to this subspace. A space with this topology will be said to be "topologized algebraically." The continuity condition on A is then easily seen to be vacuous.
Example 1.1. Let M be a finite-dimensional real vector space, and M* its dual, i.e., the space of all linear functionals on M. Let L denote the direct sum M[direct sum]M* and let A denote the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
for arbitrary x[direct sum]f and x'[direct sum]f' in L. Then it is easily verified that (L, A) is a symplectic vector space; it will be called the symplectic vector space built fromM. More generally, suppose M and N are arbitrary given real topological vector spaces, and B(x,f) is a given continuous nondegenerate bilinear form on M × N. The symplectic vector space built from (M, N, B) is defined as the space (L, A) where L = M[direct sum]N and A(x[direct sum]f, x'[direct sum]f') = B(x', f) - B(x, f').
Definition. Let (L, A) be a given symplectic vector space. A Weyl system over (L, A) is a pair (K, W) consisting of a complex Hilbert space K and a continuous map W from L to the unitary operators on K (taken as always, unless otherwise specified, in their strong operator topology) such that for all z and z' in L,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)
Equations 1.1 are known as the Weyl relations.
If H is a given complex pre-Hilbert space ("pre" signifying that completeness is not assumed), and A denotes the form A(z, z') = Imz, z', then the pair (H#, A), where H# denotes H as a real vector space with the same topology, is a symplectic vector space, a Weyl system over which is called simply a Weyl system overH. Here and below we shall follow mathematical convention and take the complex inner product <·,·> to be complex-linear in the first argument.
Example 1.2. Let L denote the space C of all complex numbers as a real two-dimensional space, and let A(z, z') = Im(zz'). Let K denote the space L2(R) of all complex-valued square-integrable functions on the real line R; here and later when the measure in euclidean space is unspecified, it is understood to be Lebesgue measure. For arbitrary z in C of the form z = x + iy, where x and y are real, let W(z) denote the operator on K,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
It is easy to check that (K, W) is Weyl system over (L,, A); it is known as the "Schrödinger system" or the "Schrödinger representation of the Weyl relations."
More generally, let H denote a finite-dimensional complex Hilbert space. Let e1, e2, ..., en denote an arbitrary orthonormal basis for H, and let H' denote the real span of the ej that is, the real subspace consisting of all real linear combinations of the ej Relative to the restriction of the given inner product <·,·> to H', H' forms a euclidean space. Let K denote L2(H'), and for arbitrary z in H, of the form z = x + iy, where x and y are in H', let W(z) denote the operator on K:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Just as in the one-dimensional case, (K, W) is easily seen to form a Weyl system over H.
The Weyl relations are a regularized form of the Heisenberg relations, which are essentially the infinitesimal form of the Weyl relations. Such an infinitesimal form, like the infinitesimal representation associated with a group representation (of which the Weyl systems are in fact a special case; cf. Problems following this section) is more effective in algebraic contexts than the global form, although the latter is more cogent for rigorous analytical purposes.
More specifically, if (K, W) is a Weyl system over a given symplectic vector space (L,A), then the map t [??] W(tz) is for any fixed z [member of] L a continuous one-parameter unitary group whose self-adjoint generator (whose existence is asserted by Stone's theorem) is denoted as φ(z). The map z [??]φ(z) from L into the selfadjoint operators on K will be called a Heisenberg system.
Theorem 1.1. Let φ denote the Heisenberg system for the Weyl system (K,W) over the symplectic vector space (L,A). Then for arbitrary vectors x and y inL,and nonzero t [member of] Rthe following conclusions can be made:
i) φ(tx) = tφ(tx)
ii) φ(tx) + φ(y) has closure φ(x + y);
iii) for arbitrary u in the dense domain D(φ(x)φ(y)) = Dφ(y)φ(x)), [(φ(x), φ(y)]u = -iA(x,y)u; and
iv) φ(x) + iφ(y) is closed.
Proof. The definition of φ makes i) clear. For the rest, we use two lemmas. Here and later the notation D(T) for an operator T denotes the domain of T.
Lemma 1.1.1. Let x and y be arbitrary inL, and suppose that u [member of] D(φ(x)). Then and W(y)u [member of] D(φ(x)) and
φ(x)W(y)u = W(y)[φ(x) + A(x, y)]u.
Proof. It follows from the Weyl relations that for arbitrary nonzero real t,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Letting t -> 0 the lemma follows.
Lemma 1.1.2. LetD'denote the set of all finite linear combinations of vectors of the form of the weak integral
∫ W(sx + ty)v F (s, t) dsdt,
where v is arbitrary inKand F is arbitrary in C∞0(R2). Then D'is dense inK, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Proof. Here and elsewhere, integrals are exended over all values of the variables of integration, unless otherwise indicated, and the notation C∞0(S) for an arbitrary manifold S denotes the set of all C∞ complex-valued functions of compact support on S.
The density of D' in K follows from the choice of a sequence {Fn} suitably approximating the Dirac measure. To show that it suffices D [subset] D(φ(x)φ(y)) it suffices to show that D' [subset] D(φ(y)) and that To show that φ(y)D [subset] D'. To show that D' [subset] D(φ(y)) is to show that if
u = ∫ W(sx + ty)v F(s, t) dsdt,
then limε->0 ε-1[W(εy) - I]u exists. Now
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
using the Weyl relations. In the integrand,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
and making a translation in t,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
letting ε -> 0 on the right side shows that the limit of the left side exists and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This shows that D' [subset] D(φ(y)) and that φ(y)D' [subset] D(φ(x)φ(y)). The proof that D' [subset] D(φ(x)) is analogous to that of D' [subset] D(φ(y)).
To see that φ(x) + φ(y) [subset] φ(x + y), note that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
so that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Now taking u in D(φ(x))[intersection]D(φ(y)), the usual argument for treating the derivative of a product shows that the right hand side of the above equation converges as t -> 0 to (φ(x) + φ(y))u. The inclusion in question follows now from Stone's theorem.
To show that φ(x + y) is the closure of φ(x) + φ(y) is thus equivalent to showing that φ(x + y) is the closure of its restriction to D(φ(x))[intersection]D(φ(y)). To prove this, recall the general criterion: a selfadjoint operator A is the closure of its restriction to a domain D if: a) D is dense; and b) eitAD [subset] D for all t [member of] R. Taking A = φ(x + y) and D = D(φ(x))[intersection]D(φ(y)), then both (a) and (b) follow from Lemma 1.1.2.
Note that by Lemma 1.1.1 for arbitrary w [member of] D(φ(x)φ(y)),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
or
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Differentiating with respect to s on the right side and setting s = 0 yields iφ(x)φ(y)w + A(y,x)w. It follows that the left side is differentiable at s = 0, and since φ(y) is closed, the limit is φ(y)φ(x)w. Thus
i[φ(y), φ(x)]w + A(y,x)w.
Conclusion iii) now follows from Lemma 1.1.2.
It also follows that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
To show that φ(x) + iφ(y) is closed, let {un} be a sequence of vectors in D(φ(x))[intersection]D(φ(y)) such that un -> u and (φ(x) + iφ(y))un -> v. Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
as m, n -> ∞. It follows in turn that so that u [member of] D(φ(x))[intersection]D(φ(y)), so that φ(x) + iφ(y) is closed, proving (iv).
(Continues...)
Excerpted from Introduction to Algebraic and Constructive Quantum Field Theory by John C. Baez, Irving E. Segal, Zhengfang Zhou. Copyright © 1992 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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