Algebraic Methods in Statistical Mechanics and Quantum Field Theory
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The four-part treatment begins with a survey of algebraic approaches to certain physical problems and the requisite tools. Succeeding chapters explore applications of the algebraic methods to representations of the CCR/CAR and quasi-local theories. Each chapter features an introduction that briefly describes specific motivations, mathematical methods, and results. Explicit proofs, chosen on the basis of their didactic value and importance in applications, appear throughout the text. An excellent text for advanced undergraduates and graduate students of mathematical physics, applied mathematics, statistical mechanics, and quantum theory of fields, this volume is also a valuable resource for theoretical chemists and biologists.
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Algebraic Methods in Statistical Mechanics and Quantum Field Theory - Dr. Gérard G. Emch
Mechanics
CHAPTER 1
General Motivation
In spite of the successes registered by quantum mechanics in the solution of physical problems involving only a finite number of degrees of freedom, it should be recognized that most of the extensions of this theory to more general situations have been plagued with difficulties, such as divergences, which physicists have had to learn to live with, short of having a better way to handle them. To lay claims to a physicist’s interest any new approach should at least demonstrate the following three points:
First, it should isolate some of the difficulties just alluded to and establish that they are indeed due to some intrinsic limitations of the usual framework. Second, the new approach should contain ordinary quantum and classical theories to the extent of being able to reproduce the successes of earlier approaches. Third, it should provide a framework leading to a proper solution of some of the traditional difficulties. The algebraic approach claims to satisfy these three requirements, the first of which is analyzed in Section 1 with the help of two illustrative models; this section concludes with some anticipations related to the third claim. The second claim is discussed in Section 2 from an axiomatic point of view.
SECTION 1. WHY NOT STAY IN FOCK SPACE?
OUTLINE. The aim of this section is to present some evidence of the necessity for an algebraic approach to physical systems involving an infinite number of degrees of freedom; these systems are encountered in both the quantum theory of fields and in the statistical mechanics of the many-body systems considered in the thermodynamical limit. We shall proceed mainly under the assumption that we can work within the familiar textbook setting; in doing so, however, we shall try to alert the reader’s attention to some of the difficulties linked to what must now be regarded as a too rigid frame of thought. The form of this section therefore is that of an analysis, leading to a diagnosis of some of the ills of the traditional theory; a cure is then suggested, the value of which is discussed in the following chapters.
We discuss first some mathematical aspects of usual quantum field theories, without which no further discussion could possibly make sense. We shall then study one representative example—namely the van Hove model—for which we can illustrate the working of the usual formalism and delineate its mathematical limitations and their physical implications. The relevance of that discussion for physics is broadened by some remarks about the BCS-model, a typical problem of statistical mechanics. This section closes with a short list of those fields in physics in which the new approach—as suggested these by two typical examples—might turn out to be useful.
a. Quantum Mechanics
Dime’s [1930] masterful exposition of the principles of quantum mechanics is, in the sequel, our prototype for what we call the traditional framework of quantum mechanics. In a different mathematical style the same ideas also form the core of von Neumann’s [1932a] account. The structure of this theory is briefly recalled in this subsection.
The observables on a physical system are identified with the self-adjoint linear operators acting on some Hilbert space .
The states of the physical system considered are identified with the density matrices , i.e., with the self-adjoint, positive, linear operators ρ of unit trace. The expectation value of an observable A on a state ρ is then computed with the formula
such that |Φ| = 1 there corresponds a state ϕ defined by
where Pϕ generated by Φ; Pϕ is then the operator defined by
or, in Dirac’s notation,
transcribes mathematically the fundamental superposition principle. Furthermore, for every such pure state ϕ the projector Pϕ can be interpreted as the observable corresponding to the statement, "the system is in the state ϕ," so that the transition probability between two states ϕ and Ψ (i.e., the probability that the system will be found in the state ϕ when we know that it is in the state Ψ) is given by
which confirms that Φ and ωΦ (where ω , Φ is referred to as the wave function corresponding to the state ϕ; Φ is clearly determined by ϕ only up to a phase ω. Finally, any density matrix can be expressed as a statistical mixture
of these states:
and
The time evolution is described by a continuous one-parameter group of unitary† operators Ut, acting either as a transformation of the states (Schrodinger picture)
hence
or as a transformation of the observables (Heisenberg picture)
These two pictures are equivalent in the sense that the evolution of the expectation values—the only measurable numbers of the theory—is given by
and, in general,
The differential form of the evolution laws is then (Stone theorem [Stone, 1932b])
The generator H of the time evolution of the vectors (the wave functions
) is identified with the Hamiltonian of the evolution of the states is called the Liouville operator.
In a similar manner we associate the momentum P with the space translations Ua, the angular momentum L, with the rotations Uθ, etc.
Finally, we mention that the observable C defined for any pair {A, B} of observables by the relation
has the following interpretation: for any state ρ provides the actual lower bound to the simultaneous observability of A and B in the sense that
this is the uncertainty principle.
This brief account of the traditional framework of quantum mechanics will suffice for our purpose in this section. The epistemological problems linked to this formalism are discussed in detail in Section 2 with the aim of helping to separate the most essential aspects of this theory from the more accidental ones.
b. Scattering Theory
The typical scattering situation can be schematized as follows. In the presence of a scattering center (whatever the latter might be) the evolution Ut is generated by a total Hamiltonian Hwould be generated by a free Hamiltonian H⁰. It seems intuitively correct to assume that far
from the scattering center the evolution is almost free,
and this is actually what the experimentalist sees: he prepares (respectively, detects) in the remote past (respectively, in the distant future) states ϕin, Ψin, … (respectively, ϕout, Ψout, …) which evolve freely. The central assumption of scattering theory is that there exists, in the mathematical description at hand, interpolating states ϕ, Ψ, …, which evolve according to the full equation of evolution (i.e., according to the prescriptions imposed by H) and which approximate ϕin, Ψin (respectively, ϕout, Ψout, …) in the remote past (respectively, in the distant future) in the sense that in the limit these states cannot be distinguished by means of the set {A}0 of observables that the experimentalist has at his disposal. Mathematically this condition is as follows:
Asymptotic condition. For each incoming state ϕin (respectively, outgoing state ϕout) there exists an interpolating state ϕ such that
REMARKS. This condition can be written equivalently as
furthermore, in actual physical situations we might wish to work with mixtures and not only with pure states; the asymptotic condition is then modified accordingly by substituting ρ for ϕ.
In the sense prescribed by the asymptotic condition the interpolating states ϕ can be considered as the limit, as t → – ∞ (respectively, t This limiting procedure formally! defines the two operators called Møller matrices:
The natural question the experimentalists want to see answered is, "What is the probability pr {Ψout ← ϕin} that the detected state Ψout will come from the prepared state ϕin?". We have
The operator S so defined
is called the collision operator, or S-matrix. The task of scattering theory is then (a) to devise a scheme for calculating the matrix elements of 5, knowing H and H⁰, and (b) to find a model H that would provide agreement between the predicted values of the matrix elements of S ) which describe the theory in the remote past (respectively, the distant future), there exists an interpolating field F(x, t) which satisfies the full equation of motion, i.e., F(x, t) = ∝t[F(x, 0)], and approximates Fin(x, t) in the remote past (respectively, Fout(x, t) in the distant future) in the following sense:
for all (physically attainable) states ϕ on the system.
Aside from the above asymptotic condition, the standard assumption is that the theory could be entirely carried out in Fock space. This standard assumption was, however, demonstrated as untenable, from a physical standpoint, by Haag [1955], who in doing so opened physicists’ eyes to the necessity for a radical departure from the traditional approach; we shall come back later (see, in particular, Section 3.1.d) to the general evidence available to support this statement. To gain some intuitive insight into the problem we nevertheless ignore, on a temporary basis, of course, Haag’s objection, proceed with the theory, apply it to a model, and watch the gathering storm.
c. Fock Space
The so-called Fock space was introduced in the quantum theory of fields by Fock [1932] and has since become the standard tool of this theory (see, for instance, Schweber [1961]). Its mathematical structure has been settled in a definitive paper by Cook [1953]. In this subsection we present briefly the essentials of the so-called Fock-space techniques in the spirit of Cook’s approach.
be the Hilbert space used in the quantum mechanical description of a single particle of species si. We first define the Hilbert space
to be used for the description of a system of N particles of species S1, S2, …, SN; this space is obtained by considering all sequences
then define a scalar product between two such sequences as
is finally the completion of this pre-Hilbert space.
In the case in which some of the particles are of the same species we have to take into account quantum statistics. In particular, if the N particles of the system considered are identical bosons, the wave function Φ(Ni.e., the Hilbert space of all admissible wave functions as follows: first, for the elements Φ(N, we define
where the sum is carried over all permutations of the N indices 1, 2, …, N. the symmetrized tensor product of N is the space of all complex-valued, square-integrable functions f(X1, X2, …, XN) (with Xk for = k 1, 2, …, N) which are totally symmetric in their arguments.
We have then, up to here, constructed the Hilbert space for the description of a system of N for the description of a system of N identical fermions can be obtained in a similar fashion, replacing the condition that Φ(N) be totally symmetric with the condition that it must be totally antisymmetric.
If we then wanted to describe a process in the course of which particles are produced, absorbed, or exchanged in one way or another, the space just constructed, accommodating only a fixed number of particles, is clearly not adequate. Fock then suggested that the Hilbert space appropriate for the second quantization is the following, which we shall refer to in the sequel as the Fock space:
In this sum ε stands for either S or Nof all complex numbers.
of all sequences
with a finite (but otherwise arbitrary) number of nonzero entries; Φ(N) is called the N-particle component with the structure of a pre-Hilbert space by the composition laws
with respect to the metric induced by this scalar product.
and the statistics is that of the bosons. For all f we define the two operators a(f) and a*(fby
We verify that for all N < ∞ and all f a(f) [resp. a(f), bounded by N ||f|| [resp. by (N + 1) ||f||]. Consequently a(f) and a(fwe conclude that a(f) and a(f), which is invariant under these mappings, so that all polynomials in a*(f) ··· a*(fN) ··· a(g1) ··· (gm), with f ··· fNg1 ··· gM In particular, the following identities are true on
for all f and g ; the last two relations are the so-called canonical commutation relations (CCR).
This is the first place in this book where we cannot avoid dealing explicitly with unbounded operators; we therefore collect a few elementary facts pertinent to their study. Let A is said to be the domain of A is said to be the range of A. Let B and AΦ = BA is said to be the restriction of B and B is said to be an extension of A . Now let A and A′ ; they are said to be adjoint to each other if (Ψ, AΦ) = (A′. A linear operator A is said to be densely defined itself. If A is densely defined, there exists a unique linear operator A*, called the adjoint of A, such that every operator A′ adjoint to A is the restriction of A. A linear operator B is said to be closed and Band BΦ = Ψ. If A is densely defined, A* is closed. A densely defined operator A is said to be symmetric and A is the restriction of A. In this case A** is a closed symmetric extension of A and is called the closure of A. A linear operator A is said to be self-adjoint . In general, a symmetric operator might have several, none, or exactly one self-adjoint extension. In particular, a symmetric operator is said to be essentially self-adjoint if its closure is self-adjoint; in this case A admits only one self-adjoint extension, namely A**. A linear operator A is said to be bounded (on its domain) if there exists a finite positive number M . When this condition is not satisfied, A is said to be unbounded. A linear operator A ] if and only if it is bounded on its domain. If A and simply speak of a bounded operator. In this case the above definitions reduce to their usual meaning; in particular, the adjoint of a bounded operator is bounded and a symmetric bounded operator is also self-adjoint.
We now come back to our study of the Fock formalism. The physical interpretation of the operators a(f) and a*(f) is checked from their very definition: if the many-body system considered is in a state represented by the wave function Φ, then a*(f)Φ is the wave function of the same system in a state that differs from the state described by Φ only in that it has one more particle, the new particle being in a state described by the one-particle wave function f; in this sense a*(f) creates a particle in wave function f and in a similar way a(fof the symmetric operator N(f) ≡ a*(f) a(f) is equal to the expectation of the number of particles of wave function f present in the state described by Φ; hence the name number operator for the (self-adjoint) closure N , where {fi | i ; it is plain that the expectation value of N the Nfor all f ; hence the state Φ0 contains no particle and is therefore interpreted as the vacuum of the theory. It is unique and satisfies the condition a(f)Φ0 = 0 for all f (which is referred to as the condition of stability can be approximated as closely as we wish by a vector obtained from Φ0 by acting on the vacuum with an appropriate polynomial in the creation operators; we refer to this property by saying that the vacuum is cyclic. Finally, if f = {fi | i left invariant by all a*(fitself; this situation is described by saying that {a*(fi) | fi ∈ f} is irreducible.
Proceeding now from the particular to the general, we define the operators a(f) and a*(fof the form
where
and any vector f , we define the vectors a(f) and a*(f) by
); then by linearity we extend a(f) and a*(frespectively; these mappings, denoted by the same symbols a(f) and a*(f.
the operators a(f) and a*(f) just constructed coincide exactly with the operators previously defined in this particular case; furthermore† the properties of these operators, previously stated in this particular case, all carry over in the general case just constructed for bosons statistics.
In Fermi statistics the same theory can be built, to parallel step by step the boson case, if we replace the symmetrizer S with the antisymmetrizer A. The only significant difference is that in the Fermi case the creation and annihilation operators satisfy the canonical anticommutation relations (CAR):
as a result of these relations the operators a(f) and a*(fand can therefore be extended by continuity to the entire .
itself.
We shall need in the sequel a definition of the second-quantized form Ω(A) of an arbitrary one-particle observable A, which will give a mathematically precise meaning to the heuristic expression
where
for an orthonormal basis {f. Such a definition has been formulated in a convenient way by Cook [1953] and we shall follow his method. Keeping in mind that we shall have to consider self-adjoint operators which are not necessarily bounded, nor have in general a completely discrete spectrum, we have first of all to define the iV-fold direct product A1 ⊗ A2 ⊗ ··· ⊗ AN be a finite sequence of Hilbert spaces and {Athe domain of Ai. Since† Awe define the linear operator A* by
we now define A1 ⊗ A2 ⊗ ··· A⊗N as the adjoint (A*)* of A. A1 ⊗ A2 ⊗ ··· ⊗ AN and in which consequently the Ai’s are one-particle operators. In particular, for each linear, densely defined, and closed operator A:
. Let us denote by A(Nsuch that Ψ(N. We then define the linear operator Ω(A) on this linear manifold by (Ω(A)Φ)(N) = A(N)Φ(NWe verify that since each A(N) is closed so is Ω(A). Furthermore, if A ), then Ω(AThe physical interpretation of this operator is now straightforward; suppose, indeed, that Φ(Nis of the form Φ1 ⊗ Φ2 ⊗ ··· ⊗ ΦN , so that A(N) is the N-particle observable corresponding to the one-particle observable Aand Ω(Aof the one-particle observable A. The symmetry requirements to be satisfied when dealing with a system of identical particles are then taken care of trivially as follows: we first notice that Ω(A, Ω(Aof the one-particle observable A. The point of the above discussion was, as announced, to give a precise mathematical meaning to Ω(A.
As an exercise left to the reader we mention the following theorem proved by Cook [1953]:
Theorem 1. Let H and A be linear, densely defined operators in and a(*)(f) the annihilation (respectively creation) operators already defined; suppose, further, that H is self-adjoint and A is closed. Then
Again this theorem is expected to hold for physical reasons; it expresses, for instance, the fact that if H is the one-particle Hamiltonian then Ω(His chosen to be the space of an (irreducible) unitary representation of the Lorentz group.
With this theorem we conclude our review of the basic aspects of the Fock-space formalism as we intend to use it.
d. The Relativistic, Free, Scalar-Meson Field
One of the main reasons for the early success of Fock-space techniques is that the space constructed in the manner described in the preceding subsection accommodates the relativistic free fields so well. For the sake of completeness and definiteness we now consider briefly the prototype provided by the scalar meson.
the Hilbert space that accommodates the irreducible representation of the Lorentz group associated with the particle we want to consider. These representations have been discussed repeatedly in the literature; in his original paper Wigner [1939] solved completely the problem of the classification of all irreducible representations of this group; the generality of his method can now be understood best from the general theory established by Mackey [1949, 1952, or 1955], who extended to a wide class of continuous groups the theory of induced representations devised by Frobenius in the case of discrete groups; abbreviated accounts of the theory, as applied to the Lorentz group, can be found in Wightman [1959, 1960, or 1962], Wigner [1962], Bargmann, Wightman, and Wigner [undated], Michel and Wightman [undated], Jauch [1959], Emch [1961 and 1963, II, Appendix], etc.; for the connection between the representations and the relativistic wave equations see Bargmann and Wigner [1948]. In view of the wide availability of these many references, we recall here only that the one-particle space for a particle of mass m and spin zero (i.e., our scalar, neutral mesonwith respect to the measure δ(p² – m²)(1/|p⁴|) dp¹ dp² dp³. This measure being concentrated on the hyperboloid (p² = mwith
and accordingly we write the relativistic scalar product
as
with
is then the self-adjoint operator defined by
such that
in the manner described in detail in the preceding subsection. On this space we define the operators a(f), a*(f; we further construct the self-adjoint operator
and its time-development
We can now use Theorem 1 to conclude that
and (denoting the time-derivative by a dot)
with, in particular at t = 0,
and furthermore
From the canonical commutation rules we now have
Four brief comments on the notation used in these relations are in order: first, these relations have to be understood as expressing the equality of the left-hand side and the operator-closure of the right-hand side: second, (f, g) denotes the scalar product with respect to the Lebesgue measure d³k; third, these equal-time
commutation relations can easily be translated to the general ones by the use of the explicit time-development of Ft(f) and Pt(f) obtained above (the validity of this last remark depends in an essential way on the fact that we are dealing here with a free field); fourth, when f and g are real, these relations reduce to
These results conclude our summary of the mathematical formalism attached to the theory of the free-scalar neutral-meson field.
The connection between this development and the usual heuristic formulation can be formally exhibited. We first enclose the system in a cubic box of volume V with the familiar discrete values of k, form an orthonormal basis of eigenfunctions of the one-particle Hamiltonian. We then consider
and
We can verify formally that for real f
which is actually the only mathematically meaningful definition of F(x); the formal commutation relations
involving the Jordan-Pauli invariant functions can be derived
formally from the commutation relations established above for the smeared
fields F(f) and P(f), the latter being the only physically sensible quantities to appear in the theory. In the same formal sense we obtain
and the Klein-Gordon equation
Our intent in the sequel is to carry over the analysis without any reference to these formal quantities, and to use only the well-defined operators a(f), a*(f), F(f), P(f
e. A Prototype for Quantum Field Theory: The van Hove Model
We have now gained enough knowledge about the Fock-space formalism for free fields to understand why this formalism is actually not sufficient for a general description of interacting fields. One of the most striking—and simplest—counterexamples to have been proposed as concrete evidence of the latter statement is the van Hove model.
To various degrees of mathematical rigor and physical insight, the properties of this model have been discussed by van Hove [1951, 1952], Friedrichs [1953, Part III], Schweber [1961, Section 12a], Kato [1961], Cook [1961], Segal [1963, Chapter V; see also other references cited there], Greenberg and Schweber [1958], and Guenin and Velo [1968]. Actually, the role of this model as an archetype can be traced back to the textbook by Wentzel [1943, §7] and its physical motivation, to the Yukawa theory of nuclear forces [1935]; it might also be mentioned that the method used by van Hove presents strong analogies with that used by Bloch and Nordsieck [1937] in their discussion of the infrared
divergences in quantum electrodynamics.
The model is a caricature of the nuclear interaction, drawn in a way that emphasizes the influence of the nucleons on the meson field. Specifically, we are considering a neutral scalar meson field in interaction with classical sources,
the latter mimicking recoilless nucleons. This downgrading of the role of the nucleons—the fact that their energy is momentum independent—is one of the essential simplifying features of the model responsible for its exact solutility.
The classical wave equation for a field F(x) interacting with a source distribution ρ(x),
suggests that the time evolution of its quantum analog F is generated by the Hamiltonian
where H0 is the free Hamiltonian of the meson field F(f) already discussed. Formally, this Hamiltonian corresponds to the heuristic form
is the Fourier transform of ρ(x).
and verify that F(ρ), hence Hwe have
is bounded below by mwe have
Consequently a(ρ) a*(ρ, and so, therefore, is F(ρ); furthermore we have
hence
This inequality enables us to use the following theorem† to conclude that H .
Theorem 2. Let A be self-adjoint on the linear manifold of a Hilbert space and let B be symmetric on such that
for all Ψ in and some constants a and b with 0 < a < 1 and Then (A + B) is self-adjoint on .
Hence H if we can find two constants a and b that satisfy the conditions of the theorem and such that
This inequality will be satisfied if we can find a and b such that
which is satisfied in turn if for all N
We can always satisfy this last inequality with a < 1 by taking
Consequently H .
Our next task is to establish the relation between the spectrum of H0 and the spectrum of H to determine how the energy of the free field F has been modified by the introduction of its interaction with the source distribution ρ To achieve this aim we notice that straightforward application of techniques similar to those just used allows us to show that P(f, hence generates a continuous one-parameter group of unitary operators Vλ(f) ≡ exp [—iP(f)λinto itself. On writing
we further see that
i.e.,
with
We mention in passing that to derive this result we used, in an essential way, the fact that the one-particle Hamiltonian is bounded below (i.e., by m > 0); Cook [1961], using a more sophisticated technique, obtained the following modification of this result, valid even if our particular assumptions were not satisfied: For any f ) of the one-particle observable H the closure is unitarily equivalent to Ω(H), with the similarity operator exp [ip(f)], where p(f) is i
We then have
is the Fourier transform of the Yukawa potential
we have
where
Since the spectrum of a self-adjoint operator is invariant under unitary transformation of this operator, we conclude from the last two relations that the result of the interaction between the meson field F and the source distribution ρ is to shift the energy (of the field) by a finite constant W; furthermore, this constant is equal to the contribution that we would obtain from a model in which the sources interacted among themselves via a Yukawa potential. Physically, this result is well known; it expresses the old idea that the nuclear forces are mediated by the meson field, and the above derivation only shows, for the moment, that this statement can be made mathematically rigorous.
We notice further that subtracting the constant W from the total Hamil-tonian would not affect the time-evolution of the expectation value of any observable. We can, therefore, if we so wish, replace in any of the equations of motion relevant to our purpose the total Hamiltonian H with the renormalized Hamiltonian
Ĥ defined by
which is then unitarily equivalent to the free Hamiltonian H0. Whenever the latter circumstance is encountered in scattering theory we conclude immediately that the S-matrix is I. Before doing so here we have to see whether we can actually speak of a scattering situation in our model. Specifically, we have to determine whether our asymptotic condition, as we formulated it in the beginning of this section, can be satisfied. The candidates for the asymptotic free fields Fin and Fout are obviously the free field
whereas we submit that
is the correct interpolating field. Since Ft(f) = F(ft), where ft stands for eiH0(1)tf, we have
a relation characteristic of a quasi-free field. Furthermore, since VĤV − 1 = H0,
is governed, as required in the formulation of the asymptotic condition, by the total (renormalized) Hamiltonian. Finally, from the commutation relations between F(f) and P(g) we conclude that
where
hence, by replacing f with f1 in the above expression we have
The asymptotic condition
then reduces to the condition
which is always satisfied for f and ρ . in Mathematically the latter conclusion is easily reached by writing this scalar product in kvanishing at infinity and continuous in the sup-norm). Physically, the fact that this limit vanishes is related to the well-known spreading of the free wave packet.
has been obtained, and the asymptotic fields Fin and Fout are identical so that the S-matrix is actually I for all physical purposes; incidentally, we also saw explicitly that the interpolating field is quasi-free (in the sense specified above).
Let us now denote by Ψ the bare vacuum for the free-field, i.e., the vector in Fock space satisfying
(which we refer to as the physical or dressed vacuum) defined by
satisfies the relations
where â(f. We can now introduce the explicit form of V to get
where
of all polynomials in the â and â*, followed by the norm closure. This space would coincide with our original Fock space, since the dressed field is unitarily equivalent to the bare field.
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