Second Quantization and the Spectral Action

We show that by incorporating chemical potentials one can extend the formalism of spectral action principle to Bosonic second quantization. In fact we show that the von Neumann entropy, the average energy, and the negative free energy of the state defined by the Bosonic, or Fermionic, grand partition function can be expressed as spectral actions, and all spectral action coefficients can be given in terms of the modified Bessel functions. In the Fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $\mu$ approaches to $0,$ can be expressed in terms of the Riemann zeta function. This recovers a recent result of Chamseddine-Connes-van Suijlekom.


Introduction
The spectral action principle of Connes and Chamseddine was originally developed mainly to give a conceptual and geometric formulation of the standard model of particle physics [2]. The spectral action can be defined for spectral triples (A, H, D), even when the algebra A is not commutative. An interesting feature here is the additivity of the spectral action with respect to the direct sum of spectral triples. Conversely, one can wonder whether a given additive functional on spectral triples is obtained via an spectral action.
In a recent paper [3], Chamseddine, Connes, and van Suijlekom have shown that the von Neumann entropy of the Gibbs state naturally defined by a Fermionic second quantization of a spectral triple is in fact spectral and they find a universal function that defines the spectral action.
In this paper we show that by incorporating chemical potentials one can extend the formalism of spectral action principle to both Bosonic and Fermionic second quantization. In fact we show that the von Neumann entropy, the average energy, and the negative free energy of the thermal equilibrium state defined by the Bosonic, or Fermionic, grand partition function, with a given chemical potential, can be expressed as spectral actions. We show that all spectral action coefficients can be expressed in terms of the modified Bessel functions of the second kind. In the Fermionic case, we show that the spectral action coefficients for the von Neumann entropy, in the limit when the chemical potential µ approaches to 0, can be expressed in terms of the Riemann zeta function. This recovers the recent result of Chamseddine-Connes-van Suijlekom in [3].
It should be noted that without the use of chemical potentials, the natural spectral function for the von Neumann entropy in the Bosonic case is singular at t = 0, and in fact the corresponding functional is not spectral.
In searching for a suitable expression of spectral action coefficients in all six cases studied in this paper, we were naturally led to the class of modified Bessel functions of the second kind. In Section 3 some basic properties of these functions are derived. In section 2 we recall some of the main concepts and results from the theory of second quantization. Our main results are presented in Sections 4 and 5. In this section, mainly to fix our notation and terminology, we shall recall some basic definitions and facts from the theory of second quantization in quantum statistical mechanics. We shall largely follow [1].

Fock space and second quantization
In this section we shall first recall the definition of the Fock space F(H) of a Hilbert space H, and the correspondding Fermionic Fock space F − (H) and the Bosonic Fock space F + (H) [1]. Here we will regard F ± (H) as subspaces of F(H), although one can also treat them as the quotient spaces of F(H) instead. After that we shall recall the procedure of second quantization.
Let H be a complex Hilbert space. We denote by H n = H ⊗ H ⊗ · · · ⊗ H the n-fold tensor product of H with itself when n > 0, and let H 0 = C. The Fock space F(H) is the completion of the pre-Hilbert space n≥0 H n . Define the projection operators P ± on H n by for all f 1 , ..., f n ∈ H. Since P ± are bounded operators with norm 1 on n≥0 H n , they can be extended by continuity to bounded projection operators on the Fock space F(H). The Bosonic Fock space F + (H) and the Fermionic Fock space F − (H) are then defined by F ± (H) = P ± (F(H)). The corresponding n-particle subspaces H n ± are defined by H n ± = P ± H n . The structure of the Fock space allows us to amplify an operator on H to the whole Bose/Fermi Fock spaces F ± (H). This procedure is commonly referred to as second quantization.
Let H be a self-adjoint operator on H with domain D(H). We define H n on H n ± by H n is essentially self-adjoint, and the selfadjoint closure of this direct sum operator is called the second quantization of the operator H and it is denoted by dΓ(H). Namely, In particular, let H = ½ be the identity operator. Then we have where N is the number operator on F ± (H), whose domain is defined by and for any ψ ∈ D(N ) For a unitary operator U on H, first we define U n on H n ± by and then extend it to the whole Fock space. We denote this extension by Γ(U ), called the second quantization of the unitary operator U , It is worth noticing that here Γ(U ) is also a unitaty operator on F ± (H). Also, if U t = e itH is a strongly continuous one-parameter unitary group acting on H, then Γ(U t ) = e itdΓ(H) on the Fock spaces F ± (H).
If H is a self-adjoint Hamiltonian operator on the one-particle Hilbert space H, then the dynamics of the ideal Bose gas and the ideal Fermi gas are described by the Schrödinger equation and the evolution of a bounded observable A ∈ B(F ± (H)) is given by conjugation as Next we shall introduce the Gibbs grand canonical equilibrium state ω of a particle system at inverse temperature β ∈ R, and with chemical potential µ ∈ R. Let be the modified Hamiltonian. Then ω is defined by Here we assume the operator e −βKµ is a trace-class operator.
If we have two one-particle spaces H 1 and H 2 , and self-adjoint operators When the operators e −dΓ(H i ) are positive trace-class operators for i = 1, 2, then

CAR and CCR algebras
Both of the CAR and CCR algebras are constructed with the help of creation and annihilation operators. Because of that, we shall recall the definitions of annihilation and creation operators first.
Let H be a complex Hilbert space. For each f ∈ H, we define the annihilation operator a(f ), and the creation operator a * (f ) acting on the Fock space F(H) by initially setting a(f )ψ (0) = 0, a * (f )ψ (0) = f , for all f ∈ H, and Here ψ (0) = 1 ∈ C. One can see that the maps f → a(f ) are anti-linear while the maps f → a * (f ) are linear. Also, one can show that a(f ) and a * (f ) have well-defined extensions to D(N 1/2 ), the domain of the operator N 1/2 . Moreover, we have that a * (f ) is the adjoint of a(f ); namely, for any φ, ψ ∈ D(N 1/2 ), one has We can then define the annihilation operators a ± (f ) and the creation operators a * ± (f ) on the Fermi/Bose Fock spaces F ± (H) by Moreover, since the annihilation operator a(f ) keeps the subspaces F ± (H) invariant, we have a ± (f ) = a(f )P ± , a * ± (f ) = P ± a * (f ).
The first relations are called the canonical anti-commutation relations (CAR), and the second relations are called the canonical commutation relations (CCR). Roughly speaking, the CAR algebra is the algebra generated by the annihilation operators a − (f ) and creation operators a * − (f ). In fact, we have the following proposition [1]:  Therefore both a − (f ) and a * − (g) have bounded extensions on F − (H).
Definition We call the subalgebra of B(F − (H)) generated by a − (f ), a * − (g) and ½ the CAR algebra and denote it by CAR(H).
Although the CCR rules looks very similar to the CAR rules, however, one can not simply mimic the previous definition of CAR algebras to deduce the definition of CCR algebras. The reason is that the annihilation operators a + (f ) and the creation operators a * + (g) are not bounded operators on F + (H).
First we introduce the set of operators {Φ(f ), f ∈ H} by Since the map f → a + (f ) is anti-linear, and f → a * + (f ) is linear, then Thus it suffices to examine the set of operators {Φ(f ), f ∈ H}. Let F + (H) = P + n≥0 H n ⊆ F + (H), i.e. F + (H) contains the sequences ψ = {ψ (n) } n≥0 which have only a finite number of nonvanishing components.
Since for each f ∈ H, Φ(f ) is essentially self-adjoint on F + (H), Φ(f ) can be extended to a self-adjoint operator, we still use Φ(f ) to denote the selfadjoint operator We have the following proposition [1]: Let CCR(H) denote the algebra generated by {W (f ), f ∈ H}. It follows that and (2) For each pair f, g ∈ H The operators W (f ) are called Weyl operators, and the algebra CCR(H) is called the CCR algabra of H.

Gibbs states
Let K µ denote the modified Hamiltonian operator In the Fermionic case, we can define the Gibbs state ω(A) over the CAR algebra CAR(H) by Here we assume the operator e −βKµ is a trace-class operator on F − (H). In fact, we have the following proposition [1]: Let H be a self-adjoint operator on the Hilbert space H and let β ∈ R. The following conditions are equivalent: (1) e −βH is trace-class on the one-particle Hilbert space H.
In the Bosonic case, we can define the Gibbs state ω(A) over the CCR algebra CCR(H) by Similarly as in the case of Fermionic Fock space F − (H), it is implicitly assumed that the operator e −βKµ is trace-class on F + (H), in fact, we have the fololowing proposition [1]: Let H be a self-adjoint operator on the one-particle Hilbert space H, let β, µ ∈ R. The following conditions are equivalent: • e −βH is trace-class on the one-particle Hilbert space H and β(H − µ½) > 0, • e −βdΓ(H−µ½) is trace-class on the Bosonic Fock space F + (H).

Entropy and energy
Let (A, H, D) be a spectral triple. We can construct the Bosonic and Fermionic Fock spaces Suppose the operator e −dΓDµ is a trace-class operator on F + (H), or on F − (H). Then we can define the density matrix In this section, we will show that when the operator e −Dµ is trace class on H, the von Neumann entropy, the average energy, as well as the negative free energy of ρ can be expressed as spectral actions for the spectral triple (A, H, D).
First let us briefly recall the von Neumann entropy and the energy. Consider a density matrix ρ on a Hilbert space H, i.e. ρ is a positive trace-class operator with Tr(ρ) = 1. Its von Neumann entropy is defined to be S(ρ) := −Tr(ρ log ρ).
Consider an observable, that is a self-adjoint operator H : H → H, and let ρ = 1 Z exp(−βH) be a thermal density matrix, at some inverse temperature β. Here Z = Tr(exp(−βH)) is the canonical partition function. Then the average energy H = Tr(ρH) is given by and the free energy −F (ρ) is defined by It is easy to see that In a given spectral triple (A, H, D), the operator e −dΓDµ is well-defined on both F + (H) and F − (H).
According to the Proposition 2.4, the operator e −dΓDµ is trace-class on F − (H) if and only if the operator e −Dµ is trace-class on H. Thus suppose e −Dµ is trace-class on H. Then we can define a density matrix on F − (H). The map D → S(ρ(dΓD µ )) gives rise to a spectral action, and this spectal action is an additive functional on spectral triples. In fact, suppose D = S ⊕ T is an orthogonal decomposition, then and since we have the entropy thus the map D → S(ρ(dΓD µ )) gives rise to a well-defined spectral action. Now for a given chemical potential µ, the map D → dΓD µ gives us a spectral action as well. According to Lemma 2.1, this action is additive. For simplicity, we take the inverse temperature β = 1 here.

Modified Bessel functions of the second kind
The modified Bessel functions {I ν (z), K ν (z)} are the solutions of the modified Bessel's equation The right-hand side of (3) should be determined by taking the limit when ν is an integer. The function I ν (z) is called the modified Bessel function of the first kind, and K ν (z) the modified Bessel function of the second kind. We shall introduce some basic properties of the modified Bessel function of the second kind. For more detail, one can check the references [7,6,4].
where γ is Euler's constant. When z ր ∞, one has Lemma 3.4. One has the integral representation formula of the function K ν (z): Lemma 3.5. Let K ν (z) be the modified Bessel functions of the second kind. Then one has [4, 8.486] Lemma 3.6. When ν > − 1 2 , a > 0, and x > 0, we have the integral formula [4,8.432] Using Lemma 3.6, we obtain the following Lemma: and where ψ ν,a (t) = 1 and ψ ν,a , φ ν,a denote the corresponding Fourier transforms of ψ ν,a and φ ν,a . Namely, Proof. According to Lemma 3.6, one has and then changing the variable t → 2πt, one can get formulae (8) and (9).
From Lemma 3.7, one can easily deduce the following lemma: and

Poisson summation and asymptotic expansions
To continue, we need the following version of the Poisson's summation formula: Lemma 3.9 (Poisson's summation formula [5]). If a function f (x) is integrable, tends to zero at infinity, and By this lemma we can deduce the following asymptotic expansion formulae [5]: Lemma 3.10. When a → 0 + , we have the following asymptotic expansions where γ is Euler's constant.
Proof. Let us consider the formula (14) first. Let By the equation (8), we have applying Lemma 3.9 to (16) and (17), Thus we get the asymptotic formula (14). If we replace a by 2a in formula (14), we get Thus we proved (12).
Remark This is consistent with the formulae given in [4, 8.526], where if we take t = 0, when x → 0 + , we can get the formulae (12) and (14) then.

The von Neumann entropy in the Fermionic second quantization
In the Fermionic Fock space, the von Neumann entropy of ρ is given by It is worth noticing that we can still define D µ for a general spectral triple, and when µ < 0, the difference between D and D µ is which is a compact operator. Thus D µ here plays the role of a fluctuation of D, even though there is no * −algebra here. Let Notice that when µ = 0, we get the same function h(x) as in [3]. The derivative of h µ (x) is According to [3], Thus we get the expansion of h µ (x): Also, according to Proposition 4.4 in [3], Thus whereg µ (t) := e µtg (t). Now we want to compute the moments of the function h µ (x), that is the integral To this end, one can first compute the two integrals separately, and then sum them up.
Lemma 4.1. We have the integral formula: Proof. According to Lemma 3.4, one has the integral formula: and using (6) and (7) From which we get the formula (24).

Lemma 4.2.
We have the following integral formula: Proof. Taking the derivative with respect to z on both sides of the formula (24), one has Using (6), one has Now substituting (27) into (26), finally we get the desired formula (25).

Lemma 4.3. When ν > −1, one has
Proof. Notice that Consider the integral and substitute x by y = x 2 − µ to get: Thus using Lemma 4.1, one has Using (30) and (29), then finally we get formula (28).

Proof. Using Propositions 4.3 and 4.4, we have
By applying (4) to this equation, we get the integral formula (32). For the second statement, we use the asymptotics and the Legendre duplication formula for the gamma function in (33), we get which is the same as [3, Lemma 4.5].
We denote the a−th order spectral action coefficient of h µ ( √ x) by γ µ (a); namely, It is clear that for a fixed chemical potential µ < 0, the equation (36) is an entire function with respect to a ∈ C. According to the Lemma 4.5, we can deduce that when the order a < 0, the coefficient of t a in the heat expansion is Now we show that for any fixed chemical potential µ < 0, the function (39) is an entire function with respect to a ∈ C , so that the function (39) can give rise to spectral action coefficients for any order a. Proposition 4.6. For any fixed chemical potential µ < 0, the function (39) is an entire function in a ∈ C. Hence we have the formula for all a.
Proof. We only need to show that the series is an entire function in a ∈ C. In fact, using the integral expression for the Bessel function K ν (z) [4,8.432], we have or Re(z) = 0 and ν = 0.
We see that for a fixed z > 0 the function K ν (z) is an entire function with respect to ν ∈ C. Now we need to show that equation (41) is locally uniformly convergent. In fact, for |ν| ≤ R, Since we have the asymptotic expansion it follows that the series ∞ n=1 n R+2 K R n √ −µ is convergent. Therefore the series (41) is locally uniformly convergent, and the function (40) is an entire function. Now according to (36), γ µ (a) is an entire function, hence the function (40) gives the spectral action coefficients for all a.
Interestingly, we can express the spectral action coefficients γ µ (a) in a more concise way via the Poisson summation formula. Proposition 4.7. For any fixed chemical potential µ < 0, we have the expression for γ µ (a): .
Proof. Using Lemma 3.8, and using the Poisson summation formula, when ν ≥ − 1 2 , a > 0, we have where φ ν,a (x) = 1 ((2x+1) 2 π 2 +a 2 ) ν+ 1 2 . Since we have the equation applying the formula (43) to Proposition 4.6 we then get the equatoin (42) when a ≥ 3 2 . Now, in Proposition 4.6 we saw that γ µ (a) is an entire function. It follows that the function (42) has an analytic extension to the whole complex plane C, and therefore equation (42) is true for all a ∈ C.
Remark The second expression of γ µ (a) is in the sense of analytic continuation. Thus for example we have, Next we prove that when the chemical potential µ → 0 − , we can get the same coefficients given in [3]. We follow the same notation as in [3], and denote where ξ(z) is the Riemann ξ−function.
Theorem 4.8. For all a ∈ n 2 : n ∈ Z , when the chemical potential µ approaches to 0, we have lim Proof. Since the spectral action coefficients are given by where g(t) is given by equation (22), we obtain Summarizing the above computations, we get the following Proposition: (1) For a given chemical potential µ < 0, the coefficient of t a in the heat expansion is given by γ µ (a), where and we have the following two explicit expressions of γ µ (a): Moveover, γ µ (a) is an entire function in a ∈ C.

The average energy in the Fermionic second quantization
Now we shall consider the average energy when the one-particle Hilbert space is H = C. We denote by Z = Tr(e −βdΓDµ ) the partition function. Then According to (1), Interestingly, this is just the first part on the right-hand side of our von Neumann entropy formula (21). We denote this function by u µ (x), Now let us consider the function u 0 (x) first. Since we have the expansion and (cf. e.g. [6]) When µ < 0, using the Fubini theorem, we can exchange the infinite sum and the integral, so that Then we obtain the following expression for the Laplace transform of r µ : Therefore, the function u µ ( √ x) is a well-defined spectral action function. Notice that here we can not take the chemical potential µ = 0, since the function u 0 (x) is singular at x = 0. When a < 0, the spectral action coefficient of t a is given by Using Lemma 4.4, we can express ω µ (a) as follows: Proposition 4.10. For any fixed chemical potential µ < 0, the function ω µ (a) is given by and moreover, it can be extended to an entire function in a.
Proof. Taking any µ < 0, and using the same argument as in the proof of Proposition 4.6, we can show that ω µ (a) can be extended to an entire function as well.
Now we want to find a more explicit expression for ω µ (a) using the Poisson summation formula.
Proposition 4.11. For any fixed chemical potential µ < 0, we can express ω µ (a) as ω µ (a) = Γ(a + 1) Proof. Using (9) and applying Poisson's summation formula, we obtain, for any ν > 0 and z > 0, When a > 1 2 , we can combine the above equation with (45), and after simplification, we can deduce the equation (46). Now since ω µ (a) is an entire function, we conclude that (46) is valid in the whole complex plane. Now we want to see how the spectral action coefficients ω µ (a) behave when µ → 0 − . Proposition 4.12. When the order a ≤ 0, we have the limit When a = 1 2 , we have the asymptotic formula When a > 1 2 , we have the asymptotic approximation Proof. For a < 0, we have Applying (44), we get When a = 0, since ω µ (0) = u µ (0), we deduce that lim When a = 1 2 , using Lemma 3.10, we see that When a > 1 2 , using Proposition 4.11, we have the limit From which (48) follows.
In particular, using Proposition 4.12, we get the expansion of u 0 ( √ x) as follows:

The negative free energy in the Fermionic Fock space
Since the free energy is the difference between average energy and von Neumann entropy, in the case of Fermionic second quantization it is natural to define the spectral action function with respect to the negative free energy to be Proposition 4.13. When chemical potential µ < 0, we have the following equation: Proof. Since 4t +tµ e −tx 2 dt. Now since µ < 0, we can apply the Fubini theorem to get the equation (49).
Therefore the function v µ ( √ x) is a well-defined spectral action function when µ < 0, while is not a well-defined spectral action function since it is singular at x = 0. We denote by λ µ (a) the spectral action coefficient of v µ (x) of order a. For a < 0 we have Using an argument similar to the subsection 4.2, we obtain the following proposition. We omit the proof which is similar to the proof of Proposition 4.10, 4.11,4.12.
Proposition 4.14. For a given chemical potential µ < 0, we can get a spectral action from the negative free energy of the Fermionic second quantization, and this spectral action function is given by the function The spectral action coefficients of v µ ( √ x) are given by the following two functions: Moreover, for any fixed chemical potential µ < 0, λ µ (a) is an entire function. When the order a < 0, we have the limit When a = 0, lim µ→0 − λ µ (0) = log 2.
When a = 1 2 , we have the asymptotic expansion: When a > 1 2 , we have the asymptotic approximation: As in the case of Fermionic Fock space, we can also define the spectral actions in the case of Bosonic Fock space. Let H = C be the 1-particle Hilbert space. Then the Bosonic Fock space is The spectrum of dΓD µ is σ(dΓD µ ) = {n x 2 − µ : n = 0, 1, 2, 3, · · · }. Since the chemical potential µ < 0, we can define a density matrix ρ = e −dΓDµ Tr e −dΓDµ .

The von Neumann entropy in the Bosonic second quantization
We define a function k µ (x) by In the Bosonic Fock space case, we cannot take the chemical potential µ = 0, since the function k 0 (x) is singular at x = 0: Lemma 5.1. The function k 0 (x) is an even positive function of the variable x ∈ R\{0}, and its derivative is . Compare this to the function h 0 (x) in section 4.1, or the function h(x) in [3], where Similar to h 0 (x), we shall prove that the function k 0 ( √ x) is also given by the Laplace transform when x = 0. To prove this, we need the following lemma(compare this with Lemma 4.2 in [3]): . Proof. We use the Eisenstein series [3] in conjunction with sinh x = −i sin(ix).
Thus 1 Now since one has the equation by the Fubini theorem we have the formula when x > 0. Now we have the following lemma: The function f (t) is rapidly decreasing as t → 0 + .
Proof. Consider the theta function Let We have f (t) = g(4πt). Thus it suffices to show that g(t) is rapidly decreasing as t → 0 + . Now, using the Jacobi inversion formula, we have Since as t → 0 + , θ ′ 1 t is rapidly decreasing, g(t) is rapidly decreasing, and also the function f (t) is rapidly decreasing as t → 0 + .
Thus we have the following proposition: where Proof. According to Lemma 5.3,f (t) is rapidly decreasing as t → 0 + . Thus when x > 0, the integral on the right hand side is well-defined. We denote the integral on the right hand side of (54) byk(x). We have and since both k 0 (x) andk(x) approach to 0 when x → ∞, thus k 0 (x) =k(x).
Thus immediately we have Proposition 5.5. When the chemical potential µ < 0, for all x ∈ R, For the Bosonic Fock space, we can get similar results as in the Fermionic Fock space case. The main difference between them is that we get alternating sum from Fermionic second quantization, while we get just a sum in the Bosonic second quantization.
Lemma 5.6. When ν > −1, one has the integral formula Proof. The proof of this proposition is the same as the proof of the Lemma 4.5.
We denote by χ µ (a) the a−th order spectral action coefficient of k µ ( √ x), that is, t a e µt f (t)dt.
Similar to the Proposition 4.6 and Proposition 4.7, we have the following proposition: Then we get the expansion of k 0 ( √ x): Unlike the Fermionic second quantization, here we cannot take µ = 0, as the integral on the right-hand side of the formula (62) does not converge. This is consistent with the fact that p 0 (x) is singular at x = 0. When the order a < 0, we denote by α µ (a) the spectral action coefficient of the spectral action function p µ ( √ x), namely, Using the same argument as in section 4.2, we have and it can be extended to a holomorphic function on C. Thus this formula gives the spectral action coefficients of all orders. Moreover, we have yet another expression for α µ (a): and it can also be extended to an entire function for any fixed chemical potential µ < 0.

The negative free energy in the Bosonic second quantization
Similar to the Fermionic second quantization, we define the spectral action function with respect to the negative free energy in the Bosonic second quantization to be It is obvious that the chemical potential must be negative, µ < 0. We denote by β µ (a) the spectral action coefficients of q µ ( √ x); namely, Using the same argument as before, we deduce that where f (x) is a non-negative even smooth function which is rapidly decreasing at ±∞, and Λ is a positive number called mass scale, or cutoff. Note that f (D/Λ) is a trace-class operator. We denote by χ(x) = f ( √ x), and assume that χ(x) is given as a Laplace transform where g(s) is rapidly decreasing near 0 and ∞. We also assume that there is a heat trace expansion Tr e −tD 2 ∼ α a α t α , t → 0 + , It was proved in [2] that the spectral action has an asymptotic expansion for Λ → ∞, namely, Tr(χ(D 2 /Λ)) ∼ a α Λ −α ∞ 0 s α g(s)ds.
When α < 0, by the Mellin transform, Thus the spectral action coefficient is And when α = n is a positive integer, since (∂ x ) n (e −sx ) = (−1) n s n e −sx , we have that