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Gauging What's RealThe Conceptual Foundations of Gauge Theories$

Richard Healey

Print publication date: 2007

Print ISBN-13: 9780199287963

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780199287963.001.0001

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(p.265) Appendix E Algebraic Quantum Field Theory

(p.265) Appendix E Algebraic Quantum Field Theory

Source:
Gauging What's Real
Publisher:
Oxford University Press

This appendix shows how algebraic quantum field theory provides a clear mathematical framework within which it is possible to raise and answer questions about the relations among various representations of the states and observables of a quantum field theory. It motivates and explains the idea of an abstract Weyl algebra of field observables and points out the interpretative significance of the fact that the Stone–von Neumann theorem does not extend to its representations. It says what is meant by a Fock representation, and explains how this is related to the occupation number representation of a system of quantum particles. Much of it relies on the paper by Ruetsche (2002) and the appendix to that by Earman and Fraser (2006).

The Heisenberg relations

[ x j , p k ] = i ħ δ jk I [ x j , x k ] = [ p j , p k ] = 0
(E.1)
generalize formally to equal‐time commutation relations (ETCRs) for field systems such as the following for operators corresponding to a real classical scalar field ϕ(x, t):
[ φ ( x , t ) , π ( x , t ) ] = i ħ δ 3 ( x x ) I [ φ ( x , t ) , φ ( x , t ) ] = [ π ( x , t ) , π ( x , t ) ] = 0
(E.2)
as well as anticommutation relations for field operators acting on states of fermionic systems such as electrons and quarks. But the presence of the delta function δ3(xx ) means that these field commutators are not really well defined. To arrive at a well‐defined algebraic generalization of the Heisenberg relations it is necessary to introduce “smeared” field operators—field operators parametrized by a family of “test” functions peaked around points like (x, t) that fall off sufficiently fast away from there (perhaps restricted even to functions of compact support). This gives rise to a basic algebra of operators of the form ̂ϕ(f x, t)), ̂π(f(x, t)) for a real scalar field, with analogous generalizations for fields of other kinds (complex, vector, etc.). As appendix D explained, it is (p.266) also necessary to replace the Heisenberg form of the canonical commutation relations by a Weyl form in which all operators are bounded and can therefore be defined on all vectors in a Hilbert space on which they act. Just as the pair of vectors (a, b) defining the Weyl operator Ŵ(a, b) for a particle system serves to pick out a point in the finite‐dimensional phase space of that particle system, so also a pair of test functions (g, f) picks out a point in the infinite‐dimensional phase space of a field system. Particle Weyl operators Ŵ(a, b) therefore generalize to field Weyl operators Ŵ(g, f).

Now on the classical phase space for a field theory like that of the Klein–Gordon field there is a so‐called symplectic form σ(f, g) that generalizes the form (a.db.c) on the phase space of a classical particle system. The multiplication rule 7.8 accordingly generalizes to

W ( g 1 , f 1 ) W ( g 2 , f 2 ) = W ( g 1 + g 2 , f 1 + f 2 ) exp ( i σ ( f , g ) / 2 )
(E.3)
which specifies a so‐called abstract Weyl algebra for the Klein–Gordon field and provides the required rigorous form of the ETCR's E.2.1 The explicit expression for the symplectic form in this case is given by the following integral over a space‐like “equal‐time” hyperplane Σ
σ ( f , g ) = Σ d 3 x ( g 1 f 2 g 2 f 1 )
(E.4)

We now face the problem of characterizing the representations of the Weyl algebra specified by equation E.3. This is the analogous problem for a quantum field theory to that considered in appendix D for a quantum particle theory. The problem is now set in the context of an algebraic approach to quantum field theory, so before we continue it is appropriate to reflect on just what that amounts to.

In the algebraic approach to quantum field theory, observables are represented by an abstract algebra  of operators, and states are represented by linear functionals s on this algebra. So if Â12 are elements of , then

s ( a A 1 + b A 2 ) = as ( A 1 ) + bs ( A 2 )
(E.5)
Such a state is intended to yield the expectation value for a measurement of an arbitrary observable in .  itself is taken to be a C* algebra: a complete, normed (p.267) vector space over the complex numbers whose elements may be multiplied in such a way that ∀Â1, Â2 ∈ , ‖Â1Â2‖ ≤ ‖Â1‖ ‖Â2‖, with an involution operation * satisfying conditions modeled on those of the Hilbert space adjoint operation, plus ∀Â ∈ , ‖Â * Â‖=‖Â‖2.2 Abstract states s on  are linear functionals satisfying
s ( A * A ) 0
(E.6)
s ( I ) = 1
(E.7)

The bounded operators of a Hilbert space ℬ(𝖧) constitute one concrete realization of a C* algebra. In the context of the algebraic approach to quantum field theory, we seek a representation in some Hilbert space of an abstract C* algebra of smeared field operators with states on them. Every representation of the Weyl algebra specified by the Weyl relations E.2 will give rise to such a representation, since the Weyl algebra constitutes a C* algebra. Arepresentation of an abstract C* algebra  on a Hilbert space 𝖧 is a * ‐homomorphism π: ℬ(𝖧) of that algebra into the algebra of bounded linear operators on 𝖧, i.e. a structure‐preserving map of elements of  onto a C* algebra constituted by elements of that algebra which satisfies the condition

( φ , π ( A ) ψ ) = ( π ( A * ) φ , ψ ) for all φ , ψ H
(E.8)
Such a representation is faithful if and only if π(Â) = 0 →  = 0, and irreducible if and only if the only subspaces of the Hilbert space 𝖧 left invariant by the operators {π(Â):  ∈  are 𝖧 and the null subspace {0}. Every representation of a Weyl C* algebra is faithful. Two representations π,π of an abstract C* algebra  are unitarily equivalent if and only if there is a unitary map U:ℬ(𝖧π) → 𝒝(𝖧π) such that π(Â) = Uπ(Â)U −1 for all  ∈ .

The Stone–von Neumann theorem does not generalize to representations of field Weyl algebras like those specified by E.3. While such an algebra does possess Hilbert space representations, these are not all unitarily equivalent to one another. Indeed, there is a continuous infinity of inequivalent representations of equation E.3’s algebra.

One important kind of representation is called a Fock representation. This is related to the occupation number representation for the quantum harmonic oscillator considered in appendix D To get the idea of a Fock representation, recall the discussion of the real Klein–Gordon field in chapter 5, section 5.1.

(p.268) The general solution to the classical Klein–Gordon equation (5.1)

μ μ ϕ + m 2 ϕ = 0
(E.9)
may be expressed as
ϕ ( x , t ) = { a ( k ) e i ( k . x ω k t ) + a * ( k ) e i ( k . x ω k t ) } d 3 x
(E.10)
where ω k 2 = k 2 + m 2 corresponds to the relativistic energy–momentum relation E 2 = p 2 c 2 + m 2 c 4 with E = hωk, p = h k and here and in the rest of this appendix we have chosen units so that c = h = 1. The canonical conjugate field π(x μ) is defined by
π ( x μ ) = 𝒧 φ · ( x μ ) = φ · ( x μ )
(E.11)
where ℒ is the Klein–Gordon Lagrangian density 𝒧 = 1 2 [ ( μ φ ) ( μ φ ) m 2 φ 2 ] . On quantization, ϕ,π become operators ̂ϕ, ̂π, and the solution to the quantized Klein–Gordon equation is
ϕ ( x , t ) = { a ( k ) e i ( k . x ω k t ) + a ( k ) e i ( k . x ω k t ) } d 3 x
(E.12)
where the commutation relations for the operators â(k) and its adjoint â(k) that follow from this and equations E.2 are
[ a ( k ) , a ( k ) ] = δ 3 ( k k ) I
(E.13)
[ a ( k ) , a ( k ) ] = [ a ( k ) , a ( k ) ] = 0
(E.14)
If we define a so‐called number operator (k)≡ â (k)â(k), then these give
[ a ( k ) , N ( k ) ] = δ 3 ( k k ) a ( k )
(E.15)
[ a ( k ) , N ( k ) ] = δ 3 ( k k ) a ( k )
(E.16)
It follows that â(k),â(k) act respectively as raising and lowering operators on eigenstates |n k 〉 of the number operator with (k)|n k〉 = δ3(0)n k|n k〉:
N ( k ) a ( k ) | n k = a ( k ) [ N ( k ) + δ 3 ( 0 ) ] | n k
(E.17)
= δ 3 ( 0 ) ( n k + 1 ) a ( k ) | n k
(E.18)
(p.269) Hence â(k)|n k 〉 is an eigenstate of (k) corresponding to eigenvalue n k + 1. Similarly, â(k)|n k 〉 is an eigenstate of (k) corresponding to eigenvalue n k − 1. Provided the system has a unique state of lowest energy, by repeatedly applying the lowering operators one arrives at that ground state—the so‐called vacuum state |0 〉 —a simultaneous eigenstate of every number operator (k) with eigenvalue n k = 0. The Hamiltonian operator Ĥ for the Klein–Gordon field has the form
H = 1 2 [ π 2 + ( ϕ ) 2 + m 2 ϕ 2 ] d 3 x
(E.19)
which becomes
H = 1 2 ω k [ a ( k ) a ( k ) + a ( k ) a ( k ) ] d 3 k
(E.20)
The commutation relations for the raising and lowering operators then give
H = ω k [ a ( k ) a ( k ) + 1 2 δ 3 ( 0 ) ] d 3 k = ω k [ N ( k ) + 1 2 δ 3 ( 0 ) ] d 3 k
(E.21)
If one follows custom in ignoring as unmeasurable the infinite zero‐point energy associated with the delta function, one can therefore try to interpret the total energy of a Klein–Gordon field as consisting of the sum of the energies ωk of all its constituent quanta of momentum k. Similarly, the total momentum represented by the operator
p = 1 2 k [ a ( k ) a ( k ) + a ( k ) a ( k ) ] d 3 k
(E.22)
might be interpreted as consisting of the sum of the momenta of all its constituent quanta. A total number operator may also be defined as
N N ( k ) d 3 k
(E.23)
whose eigenvalues might indicate the total number of quanta present in the field. The vacuum state satisfies |0〉 = 0|0〉, in accordance with its interpretation as a state in which no quanta are present. Other states of the quantized Klein–Gordon field may then be built up from the vacuum state by successive applications of linear combinations of raising and lowering operators; indeed every state in the representation may be approximated to (p.270) arbitrary precision in this way. A state |n k 〉 that can be formally “created” from the vacuum state |0 〉 by application of the raising (or “creation”) operator â(k)
| n k = a ( k ) | 0
(E.24)
is a simultaneous eigenstate of and Ĥ with eigenvalues kk respectively. It is naturally thought to contain one quantum whose energy and momentum values obey the usual relativistic relation. Repeated action with this and other “creation” operators is naturally thought to result in a state containing multiple quanta of various energies and momenta, always obeying this relation. But a typical state will be a superposition of such states, with no determinate number of quanta, and no determinate energy or momentum.

The algebraic approach makes it possible to place this heuristic treatment of a Fock representation on a sounder mathematical footing, and to state precisely what counts as a Fock representation of an abstract Weyl algebra. Instead of focusing on field operators defined at each space‐time point, one considers a corresponding abstract algebra of operators which have Hilbert space representations as “smeared” fields. In a Fock representation of a Weyl algebra, a creation or annihilation operator is parametrized not by momentum, but by an element of a complex Hilbert space 𝖧1 (called, suggestively, the one‐particle Hilbert space). For all f, g ∈ 𝖧1, their commutation relations are

[ a ( f ) , a ( g ) ] = [ a ( f ) , a ( g ) ] = 0 [ a ( f ) , a ( g ) ] = ( f , g ) I
(E.25)
permitting the definition of a number operator (f) = a (f)a(f) with
[ a ( f ) , N ( g ) ] = ( f , g ) a ( f )
(E.26)
[ a ( f ) , N ( g ) ) ] = ( f , g ) a ( f )
(E.27)
and a total number operator = ∑i a (f i)a(f i) over an orthonormal basis {f i} for 𝖧1. A symmetric Fock space 𝖥(𝖧1) is built up from 𝖧1 as the infinite direct sum of symmetrized tensor products of 𝖧1 with itself: 𝖥(𝖧1) = ℂ ⊕ s(𝖧1) ⊕ s(𝖧1 ⊗ 𝖧1) ⊕ . . .. The creation and annihilation operators are defined over a common dense domain D of 𝖥(𝖧1).3 A representation of the Weyl algebra specified by E.3 is a Fock representation in 𝖥(𝖧1) if and only if there is a unique vacuum state |0 〉 in D with a(f)|0 〉 = 0 for all f ∈ 𝖧1, and D is the span of {a (f 1)a (f 2) . . . a (f n)|0 〉 }. In a Fock representation, the total number operator is a densely defined self‐adjoint operator independent of the basis used to define it with spectrum {0,1,2,. . . }. Any representation of (p.271) the Weyl algebra defined by E.3 with such a number operator is either a Fock representation or a direct sum of Fock representations.

But the Fock representation of a free quantum field like the Klein–Gordon field is only one among an infinite number of unitarily inequivalent representations of the Weyl form of the basic ECTRs. One way to get a handle on this multiplicity is to associate representations of a Weyl algebra with states defined on that algebra. An abstract state s on an abstract Weyl algebra  (with identity Î) is a map from  into real numbers satisfying

s ( A * A ) 0
(E.28)
s ( I ) = 1
(E.29)
s ( a A + b B ) = as ( A ) + bs ( B )
(E.30)
A state s is pure just in case it cannot be expressed as a linear sum of other states. A representation of  in a Hilbert space 𝖧 is a map π: → ℬ(𝖧) from  into the set ℬ(𝖧) of bounded self‐adjoint operators on 𝖧 such that the images of elements of  themselves constitute a concrete Weyl algebra under the corresponding algebraic operations on ℬ(𝖧). Since  is a C* algebra, each state s on  defines a representation πs of the operators in  by self‐adjoint operators on a Hilbert space 𝖧s, in accordance with the Gelfand–Naimark–Segal theorem:

Any abstract state s on a C* algebra  gives rise to a unique (up to unitary equivalence) faithful representation (πs,𝖧s) of  and vector Ωs ∈ 𝖧s such that

s ( A ) = < Ω s | π s ( A ) | Ω s > for A 𝒜
(E.31)
and such that the set {πs(Â)Ωs:Â ∈  is dense in 𝖧s. This representation is irreducible if s is pure.4

Each vector |ψ 〉 in the space 𝖧 of a representation of  defines an abstract state s by s(Â) = <ψ|π(Â)|ψ > , and so to any vector that represents a state in a representation of  there corresponds a unique abstract state on . But if the GNS representations of abstract states s, s are not unitarily equivalent, then s cannot be represented as a vector or density operator on 𝖧s . Since a representation π will map the elements of  into a proper subset of the set of bounded self‐adjoint operators on 𝖧, a concrete Hilbert space representation of  will contain additional candidates for physical magnitudes represented by operators in ℬ(𝖧), over and above those represented by elements of .

Notes:

(1) The Weyl algebra itself is constituted by a set of abstract operators {Â} generated from the Ŵ(g, f) satisfying E.3 as well as Ŵ* (g, f) = Ŵ(−g, − f). It is closed under complex linear combinations. The * operation satisfies (cÂ)* =cÂ* , where c is the complex conjugate of c. The algebra possesses a unique norm ‖Â‖ satisfying ‖Â* Â‖=‖Â‖2. The Weyl algebra is also closed under this norm, making it a C* algebra.

(2) Specifically, we have the following conditions:

A 1 , A 2 𝒜 : ( A 1 + A 2 ) * = A 1 * + A 2 * , ( A 1 A 2 ) * = A 2 * A 1 * Λ , A 𝒜 : ( Λ A ) * = Λ A * , ( A * ) * = A .

(3) A set of vectors in a Hilbert space is dense just in case every vector in the space is arbitrarily close in the Hilbert space norm to a member of that set.

(4) Recall that an irreducible representation is one in which the only subspaces of 𝖧s that are invariant under the operators πs(Â) are 𝖧s and the null subspace.