THE CANONICAL ANTICOMMUTATION RELATIONS

Lecture notes for Mathematics 208 William Arveson

24 November 1998

In these notes we discuss the canonical anticommutation relations, the C^{∗}- algebra associated with them (the CAR algebra), second quantization, and the construction of KMS states for so-called free Fermi gasses. We only scratch the surface. For more, I refer you to Gert Pedersen’s book C^{∗}-algebras and their automorphism groups [3] and volume 2 of Operator algebras and quantum statistical mechanics, by Ola Bratteli and Derek Robinson [2].

Two operators X, Y are said to anticommute if XY + Y X = 0. Suppose we are given two sets of self-adjoint operators p_{1}, …, p_{n}, q_{1}, …, q_{n} acting on some Hilbert space (or more generally, belonging to some C^{∗}-algebra) which satisfy the following

p_{k}p_{j} + p_{j}p_{k} = q_{k}q_{j} + q_{j}q_{k} = 2δ_{jk}1
p_{k}q_{j} + q_{j}p_{k} = 0

for all k, j. These are the canonical anticommutation relations in their self-adjoint form for a Fermionic quantum system having n degrees of freedom. Taking j = k we find that p_{k}^{2} = q_{k}^{2} = 1 (a self-adjoint unitary operator is called a reflection). Thus, we simply have an even number of reflections which mutually anticommute with each other. The above equations make sense for infinite sequences p_{1}, p_{2}, …, q_{1}, q_{2}, … and we allow that possibility as well (indeed, most of the discussion to follow will be directed primarily to infinite systems).

These relations are best reformulated in their “complex” form, by introducing the sequence of n operators

a_{k} = \frac{1}{2}(q_{k} + ip_{k}).

After a straightforward calculation one finds that

(1)  a_{k}a_{j} + a_{j}a_{k} = 0,

a_{k}^{∗}a_{j} + a_{j}a_{k}^{∗} = δ_{jk}1, (2)

for all 1 ≤ j, k ≤ n if n is finite, and for all j, k ≥ 1 otherwise. The relations (1) and (2) are called the canonical anticommutation relations (abbreviated CARs) for n degrees of freedom.

It is also a good idea to carry this a step further, and reformulate the canonical anticommutation relations in a coordinate-free way. Assuming for the moment that n is finite and that a_{1}, …, a_{n} ∈ B(H) satisfy (1) and (2) we can define a linear map a : ℂ^{n} → B(H) in the obvious way

a(z) = z_{1}a_{1} + ⋯ + z_{n}a_{n},

z denoting (z_{1}, …, z_{n}) ∈ ℂ^{n}, and we find that

(3)  a(z)a(w) + a(w)a(z) = 0,

a(w)^{∗}a(z) + a(z)a(w)^{∗} = ⟨z, w⟩ 1, (4)

where ⟨z, w⟩ denotes the usual inner product in ℂ^{n}, z_{1}w̄_{1} + ⋯ + z_{n}w̄_{n}. If n is infinite then we replace ℂ^{n} with ℓ^{2}(ℕ), define a(z) = z_{1}a_{1} + z_{2}a_{2} + … for the dense subspace of all z ∈ ℓ^{2}(ℕ) satisfying z_{k} = 0 for all but finitely many k ∈ ℕ. Equation (4) implies that the linear map a(·) is bounded and therefore extends uniquely to a linear map of ℓ^{2}(ℕ) into B(H), and the relations (3) and (4) persist for all z, w ∈ ℓ^{2}(ℕ).

We can now free ourselves of coordinates entirely by starting with a separable complex Hilbert space Z and a linear map a : Z → B(H) which satisfies (3) and (4). Such a linear map is also called a representation of the CARs, and the number of degrees of freedom is the dimension of Z. If one chooses an orthonormal basis {e_{1}, e_{2}, … } for Z and sets a_{k} = a(e_{k}) then we recover equations (1) and (2), but of course there are many ways of choosing an orthonormal basis for Z.

There are a number of interesting things that one can deduce from (3) and (4) almost immediately. For example, the following result can be proved easily and I recommend that you supply that proof.

Proposition 1. Let a : Z → B(H) be a representation of the canonical anticommutation relations. Then for every unit vector z ∈ Z the operator U = a(z) is a partial isometry such that UH and U^{∗}H are orthocomplements of each other. The four operators e_{11} = U^{∗}U, e_{12} = U^{∗}, e_{21} = U, e_{22} = UU^{∗} are a system of 2 × 2 matrix units, and ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ae_{11} + be_{12} + ce_{21} + de_{22} defines a ∗-isomorphism of M_{2}(ℂ) onto C^{∗}(U).

Proposition 1 implies that the linear map a(·) associates to every nonzero vector z ∈ Z a 2 × 2 matrix subalgebra of B(H), namely C^{∗}(a(z)). Notice that the relation between these C^{∗}-algebras for different z is quite complicated (for example, C^{∗}(a(z_{1})) neither commutes nor anticommutes with C^{∗}(a(z_{2})), even when z_{1} ⊥ z_{2}). We will sort out the nature of this relationship presently.

It is also remarkable that the linear space of operators a(Z) is actually a Hilbert space relative to the operator norm. Indeed, since each operator a(z) has the form ∥z∥U where U is a partial isometry, we have

∥a(z)∥ = ∥z∥,

which shows that the operator norm on a(Z) is given by the inner product structure of Z. We have seen this phenomenon before when studying the Cuntz C^{∗}-algebras. Some people like to call linear spaces of operators with this property operator Hilbert spaces.

We now show how to construct representations of the CARs out of more elementary data, in a way that brings out the relation between the CARs and inductive limits of matrix algebras M_{2^{n}}(ℂ), n = 1, 2, . . .. Suppose we are given a sequence A_{1}, A_{2}, …, A_{n} of C^{∗}-subalgebras of B(H), each containing the identity operator, such that the operators in A_{k} commute with the operators in A_{j} for k ≠ j, and such that each A_{k} is isomorphic to the 2 × 2 matrix algebra M_{2}(ℂ).

For each k we choose a partial isometry u_{k} ∈ A_{k} satisfying

u_{k}u_{k}^{∗} + u_{k}^{∗}u_{k} = 1.

We know that A_{k} must contain such a partial isometry because it is isomorphic to M_{2}(ℂ). Notice first that, regardless of how one chooses u_{k}, it must generate A_{k} as a C^{∗}-algebra (for example, noting that that u_{k}^{2} = 0 we may construct a set of 2 × 2 matrix units as in Proposition 1 to see that C^{∗}(u_{k}) is itself isomorphic to M_{2}(ℂ) and thus both A_{k} and its subalgebra C^{∗}(u_{k}) are four-dimensional).

Fix k = 1, …, n. Since u_{k}u_{k}^{∗} is a projection, (1 − 2u_{k}u_{k}^{∗}) is a reflection which satisfies u_{k}(1 − 2u_{k}u_{k}^{∗}) = u_{k} and (1 − 2u_{k}u_{k}^{∗})u_{k} = −u_{k}. It follows that (1 − 2u_{k}u_{k}^{∗}) anticommutes with u_{k},

(1 − 2u_{k}u_{k}^{∗})u_{k} + u_{k}(1 − 2u_{k}u_{k}^{∗}) = 0,

while of course 1 − 2u_{k}u_{k}^{∗} commutes with u_{j} for j ≠ k. Thus

v_{k} = (1 − 2u_{1}u_{1}^{∗})(1 − 2u_{2}u_{2}^{∗}) … (1 − 2u_{k}u_{k}^{∗}), k = 1, …, n,

anticommutes with u_{1}, …, u_{k} and commutes with the remaining ones u_{k+1}, …, u_{n}. The v_{k} are mutually commuting reflections.

With these relations in hand, one easily verifies that the sequence

a_{k} = u_{k}v_{k}, k = 1, 2, …, n

satisfies the CARs (1) and (2) for n degrees of freedom. Finally, we claim that

C^{∗}(a_{1}, …, a_{n}) = C^{∗}(u_{1}, …, u_{n}) = C^{∗}(A_{1} ∪ ⋯ ∪ A_{n}) ≅ M_{2^{n}}(ℂ).

Indeed, since a_{k}a_{k}^{∗} = u_{k}u_{k}^{∗} it follows that both sets a_{1}, …, a_{n} and u_{1}, …, u_{n} generate the same C^{∗}-algebra; and by the preceding remarks this is the C^{∗}-algebra generated by A_{1} ∪ ⋯ ∪ A_{n}. It only remains to show that the latter is isomorphic to M_{2^{n}}(ℂ). We have already seen that A_{k} ≅ M_{2}(ℂ), and hence we can find a set {e_{ij}(k) : 1 ≤ i, j ≤ 2} of 2 × 2 matrix units for A_{k}, k = 1, …, n. Since these 2 × 2 systems mutually commute with each other their n-fold products

e_{i_{1}j_{1}}(1)e_{i_{1}j_{2}}(2) … e_{i_{n}j_{n}}(n), 1 ≤ i_{k}, j_{k} ≤ 2, k = 1, …, n

defines a system of 2^{n} × 2^{n} matrix units which generates C^{∗}(A_{1} ∪ ⋯ ∪ A_{n}), hence all three C^{∗}-algebras are isomorphic to M_{2^{n}}(ℂ).

Now we will show that such a sequence a_{1}, …, a_{n} ∈ B(H) is equivalent to any other representation b_{1}, …, b_{n} ∈ B(K) of the CARs in the sense that there is a unique ∗-isomorphism π : C^{∗}(a_{1}, …, a_{n}) → C^{∗}(b_{1}, …, b_{n}) such that π(a_{k}) = b_{k} for every k. To see that, notice that since the two sets {b_{k}, b_{k}^{∗}} and {b_{j}, b_{j}^{∗}} mutually anticommute with each other for k ≠ j, b_{j}b_{j}^{∗} must commute with b_{k}b_{k}^{∗} for k ≠ j. Thus we can define mutually commuting reflections ṽ_{1}, … ṽ_{n} by

ṽ_{k} = (1 − 2b_{1}b_{1}^{∗})(1 − 2b_{2}b_{2}^{∗}) … (1 − 2b_{n}b_{n}^{∗})

and corresponding operators ũ_{1}, …, ũ_{n} by ũ_{k} = b_{k}ṽ_{k}. Noting that b_{k} = ũ_{k}ṽ_{k}, a calculation (essentially reversing what was done before) shows that ũ_{k} is a partial isometry satisfying ũ_{k}ũ_{k}^{∗} + ũ_{k}^{∗}ũ_{k} = 1, and ũ_{k} commutes with both ũ_{j} and ũ_{j}^{∗} for all j ≠ k. Thus we can make a 2^{n} × 2^{n} system of matrix units out of the ũ_{k} exactly as we made one out of the u_{k} above, and since now we are talking about two systems of 2^{n} × 2^{n} matrix units, there is a unique ∗-isomorphism π : C^{∗}(u_{1}, …, u_{n}) → C^{∗}(ũ_{1}, …, ũ_{n}) such that π(u_{k}) = ũ_{k} for k = 1, …, n. This π must carry a_{k} to b_{k} in view of the relations we have seen between the a_{k}, u_{k}, v_{k} and their bedfellows b_{k}, ũ_{k}, ṽ_{k}.

What we have just done implies the following assertion about uniqueness even in the case of infinitely many degrees of freedom.

Theorem A. Let a_{1}, a_{2}, ⋯ ∈ B(H) and b_{1}, b_{2}, ⋯ ∈ B(K) be two sequences of operators satisfying (1) and (2). Then there is a unique ∗-isomorphism

π : C^{∗}(a_{1}, a_{2}, …) → C^{∗}(b_{1}, b_{2}, …)

such that π(a_{k}) = b_{k} for every k = 1, 2, . . ..

proof. For each n = 1, 2, … let A_{n} = C^{∗}(a_{1}, …, a_{n}), B_{n} = C^{∗}(b_{1}, …, b_{n}). The previous argument implies that for every n there is a unique ∗-isomorphism π_{n} : A_{n} → B_{n} such that π_{n}(a_{k}) = b_{k}, k = 1, 2, …, n.

We have already seen that A_{n} is isomorphic to a full matrix algebra, hence it is simple; and being an injective morphism of C^{∗}-algebras, π must be isometric. Because of the coherence property π_{n+1} ↾_{A_{n}}= π_{n}, there is a unique isometric ∗- homomorphism of ∪A_{n} onto ∪B_{n} which provides a common extension of the π_{n}. The closure of the latter map is the desired isomorphism π of C^{∗}(a_{1}, a_{2}, …) onto C^{∗}(b_{1}, b_{2}, …). The uniqueness of π is clear.

Theorem A implies that the C^{∗}-algebra A generated by any representation of the infinite CARs is well-defined up to ∗-isomorphism. It is a separable C^{∗}-algebra with unit. It also simple, in that it is obtained as the norm closure of an increasing sequence A_{n} ⊆ A_{n+1} of (unital) C^{∗}-subalgebras such that each A_{n} is isomorphic to the full matrix algebra M_{2^{n}}(ℂ). A is called the CAR algebra (the acronym stands for the mouth-filling canonical anticommutation relations, but it is pronounced as in “car”).

It is important to reformulate Theorem A in the following coordinate-free form.

Corollary 1. Let Z_{1}, Z_{2} be two (separable, infinite dimensional) Hilbert spaces and let a : Z_{1} → B(H) and b : Z_{2} → B(K) be linear maps satisfying (3) and (4). Then for every unitary operator U : Z_{1} → Z_{2} there is a unique ∗-isomorphism α_{U} : C^{∗}(a(Z_{1})) → C^{∗}(b(Z_{2})) such that α_{U}(a(z)) = b(Uz) for every z ∈ Z_{1}.

sketch of proof. Choose any orthonormal basis e_{1}, e_{2}, … for Z_{1} and let f_{1}, f_{2}, … be the orthonormal basis for Z_{2} defined by f_{k} = Ue_{k}, k = 1, 2, . . .. Then a_{k} = a(e_{k}) ∈ B(H), b_{k} = b(f_{k}) ∈ B(K) define two representations of the infinite CARs. Theorem A provides a ∗-isomorphism α_{U} : C^{∗}(a_{1}, a_{2}, …) → C^{∗}(b_{1}, b_{2}, …) which carries a_{k} to b_{k} for every k. One verifies easily that these C^{∗}-algebra are respectively C^{∗}(a(Z_{1})) and C^{∗}(b(Z_{2})), and that the required formula for α_{U} follows after taking linear combinations and limits.

Let a : Z → B(H) satisfy (3) and (4) and let A = C^{∗}(a(Z)) be the corresponding realization of the CAR algebra. Corollary 1 implies that every unitary operator in B(Z) gives rise to a natural ∗-automorphism of A. But what is more important for quantum physics is the following observation.

Corollary 2. Let a : Z → B(H) and A be as above. Let U = {U_{t} : t ∈ ℝ} be a strongly continuous one-parameter unitary group acting on Z. Then there is a unique one-parameter group α = {α_{t} : t ∈ ℝ} of ∗-automorphisms of A satisfying

α_{t}(a(z)) = a(U_{t}z), z ∈ Z, t ∈ ℝ.

Moreover, (A, α, ℝ) is a C^{∗}-dynamical system.

proof. The only thing that is not immediate from Corollary 1 is continuity,

(5)  lim_{t→0} ∥α_{t}(X) − X∥ = 0,  X ∈ A.

But for X = a(z), z ∈ Z, we have

∥α_{t}(a(z)) − a(z)∥ = ∥a(U_{t}z) − a(z)∥ = ∥a(U_{t}z − z)∥ = ∥U_{t}z − z∥,

which tends to zero as t → 0 by strong continuity of U. Since the set of elements a(Z) generate A as a C^{∗}-algebra, (5) follows from a now-familiar argument.

Gauge group. Taking the scalar one-parameter unitary group U_{t}z = e^{it}z, we obtain a C^{∗}-dynamical system (A, γ, ℝ) by defining

γ_{t}(a(z)) = a(e^{it}z) = e^{it}a(z), z ∈ Z, t ∈ ℝ.

γ is called the gauge group of A. It is periodic with period 2π: γ_{t+2π} = γ_{t}, t ∈ ℝ.

Hamiltonians. The one-parameter unitary groups U = {U_{t} : t ∈ ℝ} ⊆ B(Z) associated with the flow of time in quantum theory usually have spectrum that is bounded below in the sense that the spectral measure P associated with U,

U_{t} = ∫_{−∞}^{∞} e^{itλ} dP (λ),

satisfies P ((−∞, λ_{0})) = 0 for some λ_{0} ∈ ℝ. In such cases we can replace U with the phase-shifted version of itself

V_{t} = e^{−itλ_{0}}U_{t},

and this new group V has nonnegative spectrum. If we choose λ_{0} carefully (by taking it to be as large as possible) then we can ensure that the closed support of the spectral measure of V a) is contained in the nonnegative reals [0, ∞) and b) contains 0. Of course, such a λ_{0} is uniquely determined.

If we write V_{t} = e^{itH}, then H is an unbounded positive operator whose spectrum contains 0. We will make further restrictions on H presently (i.e., that it has discrete spectrum and its eigenvalues grow moderately rapidly), and for such a unitary group V we will show how to write down a KMS state for the C^{∗}-dynamical system (A, α, ℝ) associated with it.

KMS states. Let (A, α, ℝ) be a C^{∗}-dynamical system. If A is unital then there are always states on A which are invariant under the action of the group of ∗- automorphisms α (see the current exercises). We want to single out a particularly important class of invariant states, called KMS states after the physicists Kubo, Martin, Schwinger. These are abstractions of the so-called Gibbs canonical ensembles to the setting of C^{∗}-dynamical systems associated with flows. KMS states are very important not only for physical applications [2], but for the basic theory of C^{∗}-dynamical systems as well (see [3]).

For (A, α, ℝ) as above, one can always find a dense ∗-subalgebra A_{0} of A whose elements are entire with respect to the action of α in the following sense: for every a ∈ A_{0} and every bounded linear functional ρ on A, the function t ∈ ℝ ↦ ρ(α_{t}(a)) can be extended to an entire function of a complex variable t. One constructs elements of A_{0} in the following way. Choose an arbitrary element x ∈ A and let f ∈ L^{1}(ℝ) be an integrable function on ℝ whose Fourier transform has compact support. It is not hard to show that the element

a = ∫_{−∞}^{∞} f(t)α_{t}(x) dt ∈ A

is entire with respect to α, and in fact one can take A_{0} to be the subalgebra of A generated by such elements a (see [2]). For every element a ∈ A_{0} the function t ∈ ℝ → α_{t}(a) can be extended uniquely to an entire function from the complex plane ℂ into A, and for t ∈ ℂ we will write α_{t}(a) for the value of this entire function at t.

Fix a positive real number β. A state ω of A is called a KMS state (for the automorphism group α at inverse temperature β) if for every pair of elements a, b ∈ A_{0} we have

(6)  ω(aα_{iβ}(b)) = ω(ba).

Notice that on the left side of (6) we have evaluated α_{t}(b) at a point t = iβ on the imaginary axis. We also point out that, when discussing KMS states, mathematicians frequently (but not always) take β = 1.

Here are two very instructive exercises about KMS states. Consider a matrix algebra A = M_{n}(ℂ), n = 2, 3, . . .. Let H be a self-adjoint operator in A and consider the C^{∗}-dynamical system (A, α, ℝ) defined by the automorphism group

α_{t}(X) = e^{itH}Xe^{−itH}, X ∈ A, t ∈ ℝ.

Exercise 1. Show that for every β > 0,

ω_{β}(X) = \frac{trace (e^{−βH}X)}{trace (e^{−βH})}, X ∈ A

is a KMS state for (A, α, ℝ) at inverse temperature β.

Notice that in this case standard power series methodology provides us with an entire function z ∈ ℂ ↦ e^{izH} without having to appeal to the subtler methods sketched in the previous paragraphs. Note too that the state ω_{β} is invariant under the action of the automorphism group α = {α_{t} : t ∈ ℝ}.

Exercise 2. Show that the state ω_{β} of exercise 1 is the only KMS state at inverse temperature β.

We now show how to construct KMS states associated with some of the the natural flows on the CAR algebra A that are defined by Corollary 2. In order to do that we must say something about the irreducible representations of A. A is perhaps the most elementary example of a separable simple C^{∗}-algebra which has uncountably many mutually inequivalent irreducible representations. Worse, we do not even know what its irreducible representations are (indeed, we can never know a complete list of mutually inequivalent irreducible representations which is “Borel measurable”, since it is known that such a list does not exist for a C^{∗}-algebra such as A [1]). This has serious consequences: if one needs to work with an irreducible representation of the infinite canonical anticommutation relations, which irreducible representation should one choose?

Fortunately, there is a “natural” irreducible representation, which we will now discuss. While it has many favorable properties and while it is possible to single out this particular representation in terms of certain natural requirements (axioms), it is still in some fundamental sense an arbitrary choice.

Antisymmetric Fock space F_{−}(Z). We start with a separable infinite dimensional Hilbert space Z. For every n = 1, 2, … let ⋀^{n} Z denote the antisymmetric subspace of the n-fold tensor product of Hilbert spaces Z^{⊗n}. ⋀^{0} Z is defined as ℂ, and F_{−}(Z) is the direct sum

F_{−}(Z) = ℂ ⊕ Z ⊕ (Z ∧ Z) ⊕ ….

For each n ≥ 2 and z_{1}, …, z_{n} ∈ Z, the wedge product z_{1} ∧z_{2} ∧⋯∧z_{n} is defined by projecting the elementary tensor z_{1} ⊗ z_{2} ⊗ ⋯ ⊗ z_{n} ∈ Z^{⊗n} onto the antisymmetric subspace ⋀^{n} Z. ⋀^{n} Z is the closed linear span of all such expressions. If we pick an orthonormal basis e_{1}, e_{2}, … for Z, then we obtain an orthonormal basis for ⋀^{n} Z by taking all products of the form

(7)  e_{i_{1}} ∧ e_{i_{2}} ∧ ⋯ ∧ e_{i_{n}},  1 ≤ i_{1} < ⋯ < i_{n}.

It is an instructive exercise to prove the following.

Proposition. For every z ∈ Z there is a unique bounded operator a(z) ∈ B(F_{−}(Z)) satisfying

a(z)z_{1} ∧ ⋯ ∧ z_{n} = z ∧ z_{1} ∧ z_{2} ∧ ⋯ ∧ z_{n}

for n = 1, 2, …, and a(z)1 = z. This defines an irreducible representation of the canonical anticommutation relations on F_{−}(Z).

Remarks on other conventions and other mores. It is appropriate to comment here on the vagaries of notation. The operators a(z) defined in the preceding proposition are called creation operators, since when z is a unit vector a(z) creates a particle in the state z. Their adjoints a(z)^{∗} are called annihilation operators. Unfortunately, this terminology is not universal. Some people (physicists are the most likely perpretrators) like to think of the letter “a” appearing in a(z) as representing “annihilation”. Correspondingly, their a(z) would be my a(z)^{∗}. If one pursues that point of view, one is lead to define the anticommutation relations in terms of an antilinear map a : z ∈ Z → a(z) ∈ B(H) rather than by a linear map as we have done. The problem arising from these other conventions are not serious, but they can be irritating for someone who is attached to linear maps.

Finally, none of this is significant for the mathematics of the situation since by the work we have already done, any representation a_{1}, a_{2}, … of the canonical anticommutation relations is isomorphic to the sequence of adjoints b_{1} = a_{1}^{∗}, b_{2} = a_{2}^{∗}, … since the b_{k} satisfy the same CARs as the a_{k}!

Second quantization. Let A ∈ B(Z) be a contraction, ∥A∥ ≤ 1. The set of contractions is closed under operator multiplication, under the adjoint operation, and it has a natural strong and weak operator topology. Multiplication is strongly continuous (on the unit ball), but the adjoint operation is not.

The operator n-fold tensor product A^{⊗n} acts naturally on Z^{⊗n}, and it leaves the antisymmetric subspace ⋀^{n} Z invariant. Thus we can define a contraction operator Γ_{−}(A) on F_{−}(Z) in a natural way

Γ_{−}(A) = 1 ⊕ A ⊕ (A ⊗ A ↾_{⋀^{2} Z}) ⊕ (A ⊗ A ⊗ A ↾_{⋀^{3} Z}) ⊕ ….

One has Γ_{−}(A)z_{1} ∧ ⋯ ∧ z_{n} = Az_{1} ∧ ⋯ ∧ Az_{n} for every n ≥ 1, and it is a simple matter to verify that Γ_{−}(AB) = Γ_{−}(A)Γ_{−}(B), Γ_{−}(A^{∗}) = Γ_{−}(A)^{∗}, Γ_{−}(1) = 1, and that A ↦ Γ_{−}(A) is strongly continuous. The mapping Γ_{−} is called second quantization (for Fermions).

In particular, if U = {U_{t} : t ∈ ℝ} is a strongly continuous one-parameter unitary group acting on the one-particle space Z, then V_{t} = Γ_{−}(U_{t}) defines a strongly continuous one-parameter unitary group acting on F_{−}(Z). More significantly, from the definition of the operator a(z) and the formulas above, we see that V_{t}a(z)V_{t}^{∗} = a(U_{t}z) = α_{t}(a(z)) for every t ∈ ℝ, z ∈ Z. It follows that the “second quantized” unitary group V implements the action of the automorphism group α of Corollary 2 on the CAR algebra A = C^{∗}(a(Z)) ⊆ B(F_{−}(Z)) in the sense that

(8)  V_{t}XV_{t}^{∗} = α_{t}(X),  X ∈ A, t ∈ ℝ.

Let us look more carefully at (8) and the relation between U_{t} and V_{t} = Γ_{−}(U_{t}). If we write H and K for the generators of U and V respectively,

U_{t} = e^{itH}, V_{t} = e^{itK}, t ∈ ℝ,

then by definition of V we have e^{itK} = Γ_{−}(e^{itH}). If one likes to think by analogy with the exponential map of Lie group theory, then one can interpret the preceding formula as K = dΓ_{−}(H), meaning that the unbounded self-adjoint operator dΓ_{−}(H) is defined as the generator of the one parameter unitary group Γ_{−}(e^{itH}) by way of

Γ_{−}(e^{itH}) = e^{itdΓ_{−}(H)}, t ∈ ℝ.

We will not have to make use of this formalism.

More significantly, suppose now that the generator H of {U_{t} : t ∈ ℝ} is positive in the sense described in the preceding pages. Letting P be the spectral measure of H, we have

U_{t} = ∫_{0}^{∞} e^{itλ} dP (λ),

and thus for every s ≥ 0 in ℝ we can define a contraction e^{−sH} ∈ B(H) by

e^{−sH} = ∫_{0}^{∞} e^{−sλ} dP (λ).

The family of operators {e^{−sH} : s ≥ 0} is in fact a strongly continuous semigroup of positive contractions in B(Z), which is formally related to the original unitary group U by e^{−sH} = U_{is}. While the latter can be made precise, we will not have to do so.

If one now second quantizes this semigroup, one would expect that Γ_{−}(e^{−sH}) should agree with e^{−sK}, that is, that K should have nonnegative spectrum, and that its associated semigroup satisfies the expected relation with e^{−sH}. This is the case, it is not hard to prove, but we omit the argument, see [3].

We now make the key hypothesis on H, namely that for some positive s_{0} > 0, e^{−s_{0}H} belongs to the trace class

(9)  trace e^{−s_{0}H} < ∞.

(9) implies that the spectrum of H is discrete, and that if we enumerate its eigenvalues (including multiplicity) in increasing order 0 ≤ λ_{1} ≤ λ_{2} ≤ … then we have

(10)  ∑_{n=1}^{∞} e^{−s_{0}λ_{n}} < ∞.

This means that the eigenvalues λ_{n} must grow with n, but slow growth is good enough. For example, the sequence λ_{n} = log(n) satisfies (10) for any s_{0} > 1.

What we require is that the quantized semigroup e^{−sK} should also be trace-class for sufficiently large s, and the following result establishes this. This remarkable result is associated with the antisymmetry of F_{−}(Z), and in fact the corresponding property fails for symmetric “Bosonic” second quantization (the symmetric second quantization of a rank-one projection is a projection of infinite rank!).

Theorem B. Let A ∈ B(H) be a positive contraction with finite trace, and let λ_{1}, λ_{2}, … be the eigenvalues of A, repeated according to multiplicity. Then

(11)  trace Γ_{−}(A) = ∏_{k=1}^{∞}(1 + λ_{k}),

and in particular, trace Γ_{−}(A) ≤ e^{trace A} < ∞.

Remarks. The term on the right of (11) is the product of the eigenvalues of the operator 1 + A, and this suggests that one can think of (11) as the formula trace Γ_{−}(A) = det(1 + A) (of course, we have not actually given a coherent definition of the determinant of an operator on an infinite dimensional Hilbert space).

Note too that since Γ_{−}(A) is a positive operator which restricts to A on the oneparticle subspace Z ⊆ F_{−}(Z), we also have the inequality trace A ≤ trace Γ_{−}(A). Conclusion: for any positive contraction A ∈ B(H) one has

trace A < ∞ ⟺ trace Γ_{−}(A) < ∞.

proof. We can find an orthonormal basis e_{1}, e_{2}, … for Z such that Ae_{k} = λ_{k}e_{k}, k = 1, 2, . . .. Let P_{n} be the projection on the span of e_{1}, …, e_{n}, and let A_{n} = AP_{n}. Then A_{n} has finite rank and A_{n} ↑ A (strongly) as n → ∞.

Fix m = 1, 2, … and consider the restriction of Γ_{−}(A_{n}) to ⋀^{m} Z. Since

{e_{i_{1}} ∧ e_{i_{2}} ∧ ⋯ ∧ e_{i_{m}} : 1 ≤ i_{1} < i_{2} < ⋯ < i_{m}}

is an orthonormal basis for ⋀^{m} Z and A_{n}e_{i} = 0 for i > n, we see that Γ_{−}(A_{n}) vanishes on ⋀^{m} Z for m > n, and for m ≤ n we have

trace Γ_{−}(A_{n}) ↾_{⋀^{m} Z} = ∑_{1≤i_{1}<···<i_{m}≤n} ⟨Ae_{i_{1}} ∧ ⋯ ∧ Ae_{i_{m}}, e_{i_{1}} ∧ ⋯ ∧ e_{i_{m}}⟩

= ∑_{1≤i_{1}<···<i_{m}≤n} λ_{i_{1}}λ_{i_{2}} … λ_{i_{m}}.

Thus

trace Γ_{−}(A_{n}) = 1 + ∑_{m=1}^{n} trace Γ_{−}(A_{n}) ↾_{⋀^{m} Z}= 1 + ∑_{m=1}^{n} ∑_{1≤i_{1}<···<i_{m}≤n} λ_{i_{1}}λ_{i_{2}} … λ_{i_{m}}
= (1 + λ_{1})(1 + λ_{2}) … (1 + λ_{n}).

Since A_{n} ↑ A strongly as n → ∞ we have Γ_{−}(A_{n}) ↑ Γ_{−}(A), and since the trace is lower semicontinuous we conclude that

trace Γ_{−}(A) = lim_{n→∞}(1 + λ_{1})(1 + λ_{2}) … (1 + λ_{n}) = ∏_{k=1}^{∞}(1 + λ_{k}).

Since 1 + λ ≤ e^{λ} for every real number λ, we can estimate the right side of the preceding formula in the obvious way

∏_{k=1}^{∞}(1 + λ_{k}) ≤ e^{λ_{1}+λ_{2}+...} ≤ e^{trace A}

to complete the proof.

Returning now to the case e^{−sK} = Γ_{−}(e^{−sH}) under discussion, let us assume that there is a β > 0 such that trace e^{−βH} < ∞. Theorem B implies that e^{−βK} is a positive trace class operator on F_{−}(Z), and hence we can define a normal state ω_{β} of B(F_{−}(Z)) by

(12)  ω_{β}(X) = \frac{trace (e^{−βK}X)}{trace (e^{−βK})},  X ∈ B(F_{−}(Z)).

ω_{β} is called the Gibbs state (at inverse temperature β). It can be shown that it is the only normal state of B(F_{−}(Z)) whose restriction to the CAR algebra A satisfies the KMS condition (at inverse temperature β), relative to the automorphism group α = {α_{t} : t ∈ ℝ} of A defined by

α_{t}(a(z)) = a(e^{itH}z), z ∈ Z, t ∈ ℝ

(the proof can be found around page 40 of [2]).

Concluding Remarks. Let us look back over what has been done. We started with a one parameter unitary group U acting on Z and formed the associated C^{∗}- dynamical system (A, α, ℝ) by way of Corollary 2. Our goal was to construct a KMS state for (A, α, ℝ). We got going on this by first picking a particular irreducible representation of A in which the action of α could be implemented spatially by a one-parameter unitary group (obtained through the second quantization procedure).

In order to proceed further, we had to make additional hypotheses on U_{t} = e^{itH}, namely that a) H is positive, and b) e^{−βH} is trace-class for some β > 0. Under these conditions, Theorem B gave us the machinery we needed to write down the Gibbs state of (12) (we did not verify the KMS condition, but it is verified in [2]).

I do not know how to prove either existence or uniqueness of KMS states for the C^{∗}-dynamical systems obtained in this way without the additional hypotheses of the preceding paragraph. More information can be found in [3].

References

1. Arveson, W., An invitation to C^{∗}-algebras, Graduate texts in mathematics, reprinted 1998, vol. 39, Springer-Verlag, 1976.

2. Bratteli, O. and Robinson, R., Operator algebras and quantum statistical mechanics 2. Equilibrium states. Models in quantum statistical sechanics, second edition, Springer-Verlag, 1996.

3. Pedersen, G. K., C^{∗}-algebras and their automorphism groups,, Academic Press, London, 1979.