arXiv:0704.1967v2  [nucl-th]  9 Oct 2007
The microcanonical ensemble of the ideal
relativistic quantum gas
F. Becattini, L. Ferroni
Università di Firenze and INFN Sezione di Firenze
Abstract
We derive the microcanonical partition function of the ideal relativistic quantum
gas of spinless bosons in a quantum field framework as an expansion over fixed mul-
tiplicities. Our calculation generalizes well known expressions in literature in that
it does not introduce any large volume approximation and it is valid at any volume.
We discuss the issues concerned with the definition of the microcanonical ensemble
for a free quantum field at volumes comparable with the Compton wavelength and
provide a consistent prescription of calculating the microcanonical partition func-
tion, which is finite at finite volume and yielding the correct thermodynamic limit.
Besides an immaterial overall factor, the obtained expression turns out to be the
same as in the non-relativistic multi-particle approach. This work is introductory to
derive the most general expression of the microcanonical partition function fixing
the maximal set of observables of the Poincaré group.
Key words:
PACS:
1 Introduction
The microcanonical ensemble of the relativistic gas is a subject which has
not received much attention in the past. The reason of scarce interest in this
problem is the peculiarity of physical applications, which have been essentially
confined within statistical approaches to hadron production and the bag model
[1]; these are indeed the only cases where involved volumes and particle num-
bers are so small that microcanonical corrections to average quantities become
relevant. Otherwise, the involved energies or volumes are so large that canoni-
cal and grand-canonical ensembles are appropriate for most practical purposes
(e.g. in relativistic heavy ion collisions). Recently [2], it has been pointed out
that the equivalence, in the thermodynamic limit, between grand-canonical,
canonical and microcanonical ensembles does not apply to fluctuations, more
http://arxiv.org/abs/0704.1967v2
in general to moments of multiplicity distributions of order > 1. Indeed, effects
of the difference between statistical ensembles might be unveiled in studying
multiplicity distributions in relativistic nuclear collisions, [3]. In view of these
phenomenological applications, it would be then desirable to have an in-depth
analysis of the microcanonical ensemble of a relativistic quantum gas.
The main difficulty in tackling this problem stems from the need of imposing a
finite volume. This is necessary to have a correct thermodynamic limit because,
being the energy E finite by construction, also V must be finite if the limit
with E/V fixed is to be taken. Strange as it may seem, a full and rigorous
treatment of the relativistic microcanonical ensemble of an ideal gas at finite
volume is still missing. In all previous works on the subject, at some point, the
large volume approximation is introduced; this is tacitly done, for instance,
considering the single-particle level density as continuous, namely replacing
sums over discrete quantum states with momentum space integrations [4]:
d3p . (1)
In a previous work [5], we have derived an expression of the microcanoni-
cal partition function of an ideal relativistic quantum gas with explicit finite-
volume corrections (see eqs. (23),(24)). However, that expression was obtained
in an essentially multi-particle first-quantization framework, which, as pointed
out in ref. [6] should be expected to become inadequate at very low volumes,
comparable with the Compton wavelength of particles. In this regime, under-
lying quantum field effects should become important and pair creation due to
localization an unavoidable effect [7]. Indeed, there are several studies of the
microcanonical ensemble of a free quantum field [8], but all of them, again, at
some point invoke a large-volume approximation. In the limit of large volumes
one obtains the same expressions of the microcanonical partition function and,
consequently, of statistical averages as in the first-quantization multi-particle
approach followed in ref. [5].
The aim of this work is to derive a general expression of the microcanonical
partition function in a full relativistic quantum field framework, valid for any
finite volume, generalizing the results obtained in [5]. We will do this for the
simplest case of an ideal gas of spinless bosons and postpone the treatment
of particles with spin to a forthcoming publication [9]. We will show that
the expression of the microcanonical partition function obtained in ref. [5]
in a non-relativistic multi-particle approach holds provided that a consistent
prescription of subtracting terms arising from field degrees of freedom outside
the considered volume is introduced.
The paper is organized as follows: in Sects. 2 and 3 we will argue about general
features of the microcanonical ensemble for a relativistic system and discuss
several issues concerning a proper definition of the microcanonical partition
function. In Sect. 4 we will cope with the further issues related to the definition
of a microcanonical ensemble for a quantum field at finite volume. Sects. 5
and 6 include the main body of this work, where the microcanonical partition
function is worked out in a quantum field theoretical framework. In Sect. 7 we
will summarize and discuss the results.
2 On the definition of the microcanonical partition function
It is well known that the fundamental tool to calculate statistical averages in
any ensemble is the partition function. For the microcanonical ensemble one
has to calculate the microcanonical partition function (MPF) which is usually
defined as the number of states with a definite value E of total energy:
states
δ(E − Estate) . (2)
For a quantum system, the MPF is the trace of the operator δ(E − Ĥ): 1
Ω = trδ(E − Ĥ) (3)
with proper normalization of the basis states. For instance, for one non-
relativistic free particle, one has to calculate the trace summing over plane
waves normalized so as to 〈p|p′〉 = δ3(p− p′):
Ω = tr(E − Ĥ) =
E − p
〈p|p〉 = 1
(2π)3
d3p δ
E − p
Thereby, one recovers the well known classical expression implying that the
MPF is the number of phase space cells with size h3 and given energy E. This
number is infinite as the volume is unbounded and it is thus impossible to
calculate a meaningful thermodynamic limit at finite energy density. Hence,
one usually considers a system confined within a finite region by modifying the
hamiltonian Ĥ with the addition of infinite potential walls, i.e. setting Ĥ ′ =
Ĥ+ V̂ where Ĥ is the actual internal hamiltonian and V̂ an external potential
implementing infinite walls. Classically, this leads to a finite Ω, namely:
(2π)3
d3p δ
E − p
1 Throughout this work quantum operators will be distinguished from ordinary
numbers by a symbol “ˆ”.
where V is the volume of the region encompassed by the potential walls. 2
Also the corresponding quantum problem can be easily solved and one has:
Ω = tr(E − Ĥ ′) ≡
E − k
where the sum runs over all wave vectors k which, for a rectangular box
with side Li and periodic boundary conditions, are labelled by three integers
(n1, n2, n3) such that ki = niπ/Li. The difficulty of the quantum expression
(6) with respect to the classical one (5) is that, for a given energy E, a set of
integers fulfilling the constraint imposed by the Dirac’s δ:
in general does not exist. Therefore, the MPF vanishes except for a discrete set
of total energies, for which it is divergent. One has a finite result only for the
integral number of states
0 dE Ω(E), that is the number of states with an
energy less than a given E ′, but this is clearly a stepwise and non-differentiable
function of E ′.
This holds for an ideal gas of any finite number of particles: strictly speaking,
the MPF cannot be defined at finite energy and volume as a continous func-
tion. Only in the thermodynamic limit E → ∞ and V → ∞ an expression like
(6) becomes meaningful because it is then possible to replace the sum over
discrete levels with a phase space integration:
cells
(2π)3
d3p . (7)
Therefore, for a truly finite quantum system, one needs a better definition
of microcanonical partition function. A definition which does not suffer from
previous drawbacks is the following:
Ω = trV δ(E − Ĥ) ≡
〈hV |δ(E − Ĥ)|hV 〉 . (8)
where Ĥ is the internal hamiltonian, without external confining potential, and
the |hV 〉’s form a complete set of normalized localized states, i.e. a complete
set of states for the wavefunctions vanishing out of the region V . It should
be stressed that these states are not a basis of the full Hilbert space because
wavefunctions which do not vanish out of V cannot be expanded onto this
basis; thence the notation trV instead of tr meaning that the trace in eq. (8)
is not a proper one. The difference between (8) and a definition like (3) is
that |hV 〉 are not eigenstates of the hamiltonian Ĥ and the right hand side of
2 We will use the same symbol V to denote both the finite region and its volume.
(8) does not reduce to a discrete sum of δ’s. In fact, this is crucial to have a
continuous function of E, unlike (6).
As an example, let us work out the definition (8) for the single free particle
confined in a rectangular box by infinite potential walls and compare it to (4).
A complete set of states for this problem is:
|k〉 =
exp(ik · x) if x ∈ V k = πnx/Lxπ̂i + ny/Ly ĵπnz/Lzk̂
0 if x /∈ V
Therefore, the MPF definition (8) implies:
〈k|δ(E − Ĥ)|k〉 =
d3p |〈k|p〉|2δ
E − p
where we have inserted a resolution of the identity by using a complete set of
states for the full Hilbert space. The sum in (10) can be calculated and yields:
|〈k|p〉|2 =
V (2π)3
d3x exp[i(k− p) · x]
d3x′ exp[ik · (x− x′)] exp[−ip · (x− x′)]
(2π)3
d3x′ δ3(x− x′)] exp[−ip · (x− x′)] = V
(2π)3
where the completeness relation in V :
exp[ik · (x− x′)] = δ3(x− x′) (12)
has been used. Thus, by using (11), the eq. (10) turns into:
(2π)3
d3p δ
E − p
that is the same expression (5) as in the classical case.
The MPF (13) is now manifestly a continuous function of E and, remarkably,
its thermodynamic limit V → ∞ is the same as the thermodynamic limit of
the “pure” quantum expression (6) (because of (7)). Since the only strict re-
quirement for a well defined MPF is to reproduce the correct thermodynamic
limit, for a gas one can choose a definition like (8) instead of Ω = trδ(E− Ĥ ′)
in (6). We emphasize again that in the passage from (6) to (8) the hamilto-
nian embodying an external confining potential is replaced with the internal
hamiltonian while, at the same time, the localized eigenstates of the former
hamiltonian are used to calculate the trace.
3 The microcanonical partition function of a relativistic system
In special relativity, the microcanonical ensemble must include momentum
conservation beside energy’s to fulfill Lorentz invariance. This means that the
MPF definition (8) should be generalized to [5]:
〈hV |δ4(P − P̂ )|hV 〉 , (14)
P being the four-momentum of the system and P̂ the four-momentum op-
erator. The MPF now being a number of states per four-momentum cell, it
is a Lorentz-invariant quantity. The calculation of the MPF is easiest in the
rest-frame of the system, where P = (M, 0) and the four-volume V u, u being
the four-velocity and V the proper volume of the system, reduces to (V,0),
according to the usual formulation of statistical relativistic thermodynamics
[10,11].
The eq. (14) can be further generalized by enforcing the conservation of not
only energy-momentum but of the maximal set of conserved quantities, i.e.
a maximal set of commuting observables built with the generators of the
Poincaré group. To achieve this, one has to replace δ4(P − P̂ ) in (14) with
a generic projector Pi over an irreducible state of the representation of the
Poincaré group [5,6], i.e:
〈hV |Pi|hV 〉 . (15)
This ensemble is still generally defined as microcanonical ensemble and (15)
microcanonical partition function.
In this work, we will confine to a microcanonical ensemble where only energy-
momentum is fixed, i.e. our projector Pi in eq. (15) will be:
Pi = δ
4(P − P̂ ) (16)
and to an ideal quantum gas, i.e. with P̂ being the free four-momentum op-
erator. In fact, it should be stressed that δ4(P − P̂ ) is not a proper projector,
because P2 = aP where a is a divergent constant. This is owing to the fact
that normalized projectors onto irreducible representations cannot be defined
for non-compact groups, such as the space-time translation group T(4). Nev-
ertheless, we will maintain this naming even for non-idempotent operators,
relaxing mathematical rigor, because it will be favorable to adopt the projec-
tion formalism.
It is worth pointing out that the definition (8) involving only the internal (free)
hamiltonian, is much more fit than (6) for a relativistic generalization. Besides
the advantage of restoring continuity in E, discussed in previous section, this
formulation can be easily extended to the full set of conservation laws with-
out major conceptual difficulty. Conversely, had one tried to generalize (6),
one should have defined a finite region and afterwards sought the observables
commuting with the hamiltonian supplemented with an external confining po-
tential. This would have not been an easy task, and, moreover, a maximal set
of observables commuting with the modified Ĥ would not, in general, define a
Poincaré algebra. This is a well known problem in the static bag model where
the translational invariance is manifestly broken and momentum is thus not
conserved. On the other hand, in the definition (15), we deal with the original
Poincaré algebra of unmodified (free) operators and enforce the localization
through the projector onto confined states.
The eq. (15) can be recast as a full trace by inserting a complete set of states
|f〉 into (15):
〈hV |
|f〉〈f |Pi|hV 〉
〈f |Pi
|hV 〉〈hV |f〉 ≡
〈f |PiPV |f〉 = tr [PiPV ] (17)
where
|hV 〉〈hV | , (18)
is, by definition, the projector onto the Hilbert subspace HV of confined states
(i.e. of wavefunctions vanishing out of V ). The formula (17) is the starting
point to carry out a calculation of the MPF at finite volume. The first thing
to do is to expand (17) as a sum of partition functions at fixed multiplicities,
i.e.:
ΩN (19)
for a single species gas and:
Ω{Nj} (20)
for a multi-species gas, where {Nj} = (N1, . . . , NK) is a set of particle mul-
tiplicities for each species j = 1, . . . , K, defining a channel. ΩN or Ω{Nj} are
obtained by summing over all possible values of kinematical variables with
fixed multiplicities. So, if |f〉 ≡ |N, {p}〉 where {p} labels the set of kinemati-
cal variables of particles in the state |f〉, ΩN reads:
〈N, {p}|PiPV |N, {p}〉 . (21)
Likewise, for a multi-species gas, the microcanonical partition function is ex-
pressed as an expansion over all possible channels:
Ω{Nj} =
〈{Nj}, {p}|PiPV |{Nj}, {p}〉 . (22)
and Ω{Nj} is defined as the microcanonical channel weight.
The microcanonical channel weights (22) have been calculated in ref. [5] with
energy-momentum conservation (i.e. using (16) in a multi-particle, first quan-
tization framework. For a single species ideal spinless gas:
d3p1 . . .d
3pN δ
ρ∈ SN
FV (pρ(n) − pn) (23)
while for a multi-species gas of K spinless bosons:
Ω{Nj} =
d3pnj
ρj∈ SNj
FV (pρj(nj) − pnj ) (24)
being N =
j Nj . In eqs. (23),(24) ρj labels permutation belonging to the
permutation group SNj and FV are Fourier integrals over the region V :
FV (p− p′) ≡
(2π)3
d3x ei(p−p
′)·x (25)
If the volume is large enough so as to allow the approximation:
FV (p− p′) =
(2π)3
d3x ei(p−p
′)·x ≃ δ3(p− p′) (26)
the microcanonical channel weights (24) can be resummed explicitely into the
microcanonical partition function according to (20) and one obtains [5]:
limε→0
(2π)4
∫ +∞−iε
−∞−iε
d3y eiP ·y exp
(2π)3
d3p log(1− e−ip·y)−1
A full analytical calculation of eq. (27) is possible only for the limiting case
of vanishing masses (e.g. microcanonical black body). For the massive case,
four-dimensional integrations cannot be worked out analytically and one has
to resort to numerical computation.
The eq. (27) was implicitely obtained in ref. [4] where the first expression
of the microcanonical partition function of a multi-species ideal relativistic
quantum gas was derived as an expansion (20) over channels, by using the
large-volume approximation (7) from the very beginning. This shows that the
approximation (26) is indeed equivalent to the (7), as also demonstrated in
ref. [5]. Noticeably, the MPF definition eq. (14) without any large volume ap-
proximation involves the appearance of Fourier integrals accounting for Bose-
Einstein and Fermi-Dirac correlations in the quantum gas, which do not show
up in the large-volume approximation enforced in ref. [4]. This approach also
allows to investigate further generalizations when the volume is so small that
relativistic quantum field effects must be taken into account.
4 Microcanonical ensemble and field theory
The calculation of the partition function (14) in a quantum field framework
brings in new difficulties with respect to the first-quantization scheme. This
problem has been approached in literature with a functional approach, in-
spired of the usual grand-canonical thermal field theory [8]. However, these
calculations aim at the limit of large volumes and are therefore insensitive to
the difficulties pertaining to the strict requirement of finite volume discussed
in detail in Sect. 2. As a result, for a free field, the derived expressions are
equivalent to the formula (27).
Instead of starting with a functional integration from the very beginning, we
calculate the microcanonical partition function of a free field by first expanding
it at fixed multiplicities like in eqs. (19),(21) (or channels, for multi-species
gas like in eq. (20)), where |N, {p}〉 are Fock space states with definite particle
multiplicity and kinematical variables {p}. To carry out this calculation, we
first need to find an expression of the microcanonical state weight:
ω ≡ 〈N, {p}|PiPV |N, {p}〉 . (28)
By using (16) and choosing |N, {p}〉 as an eigenstate of the four-momentum
operator with eigenvalue Pf =
i pi, the eq. (28) becomes:
ω = δ4(P − Pf) 〈N, {p}|PV |N, {p}〉 , (29)
To calculate ω and, by a further integration, ΩN we need to know the projector
PV . Since PV is defined as the projector onto the Hilbert subspace of localized
states, it can be easily written down in a multi-particle non-relativistic quan-
tum mechanical (NRQM) framework. As an example, for a non-relativistic
spinless single particle, it reads (see also Sect. 2):
|kV 〉〈kV | (30)
where |kV 〉 is a normalized state of the particle confined in a region V , with a
corresponding wavefunction ψkV (x) vanishing out of V . The symbol kV stands
for a set of three numbers labelling the kinematical modes of the confined
states (e.g. discrete wavevectors, or energy and angular momenta) and the set
|kV 〉 form a complete set of states for the wavefunctions vanishing out of V .
The projector (30) can be easily extended to the many-body case and:
Ñ ,{k}
|Ñ, {kV }〉〈Ñ, {kV }| . (31)
where the symbol {kV } denotes a multiple set of kinematical modes of the
confined states while Ñ is the integrated occupation number, i.e. the sum of
occupation numbers over all single-particle kinematical modes. In the NRQM
approach these numbers are simply particles multiplicities, implying:
〈N, {p}|Ñ, {kV }〉 6= 0 iff N = Ñ . (32)
To calculate the microcanonical state weight, thence the MPF, the products
〈Ñ, {kV }|N, {p}〉 can be worked out on the basis of (32) similarly to what has
been done in Sect. 2 for a single particle, yielding, for a scalar boson [5]:
〈N, {p}|PV |N, {p}〉 =
ρ∈ SN
FV (pρ(n) − pn) (33)
where ρ is a permutation of the integers 1, . . . , N and FV are Fourier inte-
grals (25) over the system region V . From the above equation the expression
of the microcanonical channel weight in eq. (23) follows. We will refer to the
expression (33) as the NRQM one, meaning that is has been obtained in this
first-quantized multi-particle NRQM framework, where, for instance, particles
and antiparticles are simply considered as different species and their contri-
butions factorize.
One could envisage that a projector like (31), written in terms of Fock space
states, could be simply carried over to the relativistic quantum field case,
where |Ñ , {kV }〉 are states of the localized problem, obtained by solving the
free field equations in a box with suitable boundary conditions. Yet, some
difficulties soon arise. First of all, whilst in NRQM the single-particle local-
ized wavefunction |kV 〉 and the free plane wave state |p〉 live in the same
Hilbert space, in quantum field theory the localized and the non-localized
problem are associated with distinct Hilbert spaces. Thus, unlike in NRQM,
it is not clear how to calculate a product like 〈Ñ, {kV }|N, {p}〉. Secondly, even
if there was a definite prescription for it, it should be expected that the inte-
grated occupation numbers of the localized problem do not coincide with actual
particle multiplicities unless the volume is infinite. To understand this point,
one should keep in mind that properly called particles arise from solutions of
the free field equations over the whole space and that the hamiltonian eigen-
states of the localized problem are conceptually different. Consequently, the
integrated particle number operators in the whole space should differ from in-
tegrated number operators within the finite region. Hence, unlike in NRQM,
a state with definite integrated occupation numbers Ñ (we purposely refrain
from calling them particle numbers) should be expected to have non vanish-
ing components on all free states with different numbers of actual particles,
namely:
|Ñ〉V = α0,Ñ |0〉+ α1,Ñ |1〉+ . . .+ αÑ ,Ñ |Ñ〉+ . . . (34)
where α
are non-trivial complex coefficients, and (32) no longer holds. Only
in the large volume limit one expects that the integrated occupation numbers
coincide with actual multiplicities and eq. (32) applies. This kind of effect is
pointed out in the introduction of Landau’s book on quantum field theory
[7]: when trying to localize an electron, an electron-positron pair unavoidably
appears, meaning that the localized single “particle” is indeed a superposition
of many true, asymptotic particle states. Another relevant manifestation of
this difference which is probably more familiar, is the Casimir effect, which is
related to the difference between the true vacuum state |0〉 and the localized
vacuum state |0〉V . Hence, all formulae derived under the approximation (32)
are asymptotic ones, valid in the limit V → ∞ but not at strictly finite volume.
Thus, one should expect significant finite-volume corrections to the eqs. (33)
and the ensuing MPF (23) in a quantum field treatment.
If we want to give an expression like 〈Ñ , {kV }|N, {p}〉 a precise meaning in a
quantum field framework, we first need to map the Hilbert space HV of the
localized problem into the Hilbert space H of the free field over the whole
space. This can be done by mapping the field eigenstates and operators of H
into HV in a natural way as:
Ψ(x)HV −→ Ψ(x)H
|ψ(x)〉HV −→ |ψ(x)〉H (35)
This allows writing linear, non-bijective, Bogoliubov relations expressing the
annihilation and creation operators of the finite region problem as a function
of those of the whole (real scalar) field (see Appendix A for derivation):
d3pF (k,p)
εk + εp
ap + F (k,−p)
εk − εp
a†p (36)
where k are triplets of numbers labelling kinematical modes, just like the
aforementioned kV , εk is the associated energy, εp =
p2 +m2 and:
F (k,p) =
(2π)3
d3xu∗k(x) e
ip·x (37)
uk being a complete set of orthonormal wavefunctions for the finite region.
A remarkable feature of relativistic quantum fields is that, unlike in NRQM,
the localized annihilation operators have non-vanishing components onto the
creation operators in the whole space, as shown by (36). This confirms our ex-
pectation that a localized state with definite integrated occupation numbers is
a non-trivial linear combination of states with different particle multiplicities.
Expectedly, as the volume increases, these components become smaller and in
the infinite volume limit one recovers ak = ap (see Appendix A).
Starting from the Bogoliubov relations (36), it should be possible, in principle,
to calculate the coefficients in eq. (34), thence the microcanonical state weight
(29) by using the expansion (31). In fact, we do not really need to do that.
It is more advantageous, as pointed out in ref. [6], to write the projector PV
in terms of field states rather than occupation numbers of field modes within
the finite region. Indeed, in the general definition in eq. (17):
|hV 〉〈hV | (38)
the states |hV 〉 are a complete set of states of the Hilbert space of the localized
problem HV , where the degrees of freedom are values of the field in each
point of the region V , i.e. {ψ(x)} |x ∈ V . Therefore, the above projector is a
resolution of the identity of the localized problem and can be written as (for
a real scalar field):
Dψ|ψ〉〈ψ| (39)
where |ψ〉 ≡ ⊗x|ψ(x)〉 and Dψ is the functional measure; the index V means
that the functional integration must be performed over the field degrees of
freedom in the region V , that is Dψ = ∏x∈V dψ(x). The normalization of
the states is chosen so as to 〈ψ(x)|ψ′(x)〉 = δ(ψ(x) − ψ′(x)) to ensure the
idempotency of PV . If we now want to give expressions like:
〈{Nj}, {p}|PV |{Nj}, {p}〉 (40)
a clear meaning, we should find a way of completing the tensor product in the
projector (39) with the field states outside the region V such a way the scalar
product can be performed unambiguously.
Unfortunately the answer to this question is not unique and the projector
can be extended to H in infinitely many ways. What is important is that the
result of the calculation is independent of how the projector has been extended.
Thus, at the end of the calculation, one has to check whether spurious terms
appear, possibly divergent, depending explicitely on the chosen state of the
field outside V and these terms must be subtracted away. In general, all terms
depending on the degrees of freedom of the field out of V must be dropped
from the final result.
In this work, we will extend the projector with eigenstates of the field, where
the field function ψ(x) is some arbitrary function outside the region V . Thus,
the projector PV (39) is mapped to:
Dψ|ψ〉〈ψ| |ψ〉 ≡ ⊗x∈V |ψ(x)〉 ⊗x/∈V |ψ(x)〉 (41)
where the index V still implies Dψ = ∏x∈V dψ(x). We will see that, with the
definition (41), spurious terms depending on the degrees of freedom outside
V do arise indeed, but that they can be subtracted “by hand” in a consistent
5 Single particle channel
We will start calculating the expectation value 〈p|PV |p〉 of a single parti-
cle channel in the simple cases of neutral and charged scalar fields. This is
preparatory to the general case of multiparticle states in Sect. 6.
5.1 Neutral scalar field
We consider a gas made of one type of spinless boson described by the free
real scalar field 3 (in Schrödinger representation):
Ψ(x) =
a(p) eip·x+a†(p) e−ip·x
where ε ≡
p2 +m2 is the energy, p is the modulus of the three-momentum
and the normalization has been chosen so as to have the following commutation
rule between annihilation and creation operators:
[a(p), a†(p′)] = δ3(p− p′) . (43)
We start writing the one-particle Fock state |p〉 in terms of creation and an-
nihilation operators acting on the vacuum:
〈p|PV |p〉 = 〈0|a(p) PV a†(p)|0〉 . (44)
3 Henceforth, the capital letter Ψ will denote field operators while for field functions
we will use the small letter ψ.
Since PV is defined, according to (41) as a functional integral of eigenvectors
of the field operator Ψ, it is convenient to express creation and annihilation
operators in terms of the field operators. We shall use following expressions
which are the most appropriate for our task:
〈0|a(p)= 〈0| 1
d3x e−ip·x
2ε Ψ(x)
a†(p)|0〉= 1
d3x eip·x
2ε Ψ(x) |0〉 . (45)
These expressions can be easily checked by plugging, on the right hand side,
the field operators in (42). By using eq. (45) in (44):
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x′ eip·(x−x
′) 2ε 〈0|Ψ(x′)PVΨ(x)|0〉 (46)
It can be easily checked now that the rightmost factor in the above equation
turns out to be (by using the definition (41)):
〈0|Ψ(x′)PVΨ(x)|0〉 =
Dψ |〈ψ|0〉|2ψ(x′)ψ(x) . (47)
where ψ(x) and ψ(x′) are field functions or the eigenvalues of the field operator
relevant to the state |ψ〉, that is:
Ψ(x)|ψ〉 = |ψ〉ψ(x) . (48)
It is possible to find a solution of the functional integral (47) by first consid-
ering the infinite volume limit, when the projector PV reduces to the identity.
In this limiting case, the functional integral in (41) is now performed over
all possible field functions and eq. (41) becomes a resolution of the identity;
〈0|Ψ(x′)Ψ(x)|0〉 is just the two-point correlation function that we write, ac-
cording to (47):
〈0|Ψ(x′)Ψ(x)|0〉 =
Dψ |〈ψ|0〉|2ψ(x′)ψ(x) (49)
The product 〈ψ|0〉 is known as the vacuum functional and reads [12], for a
scalar neutral field:
〈ψ|0〉 = N exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
where N is a field-independent normalization factor, which is irrelevant for
our purposes. The function K(x′ −x) is called kernel and fulfills the equation
[12]: ∫
d3x′ e−ip·x
K(x′ − x) = 2ε e−ip·x (51)
whose solution is:
K(x′ − x) = 1
(2π)3
d3p e−ip·(x
′−x) 2ε . (52)
The functional integral (49) is therefore a gaussian integral and can be solved
by using the known formulae for multiple gaussian integrals of real vari-
ables [12]:
I2N =
ψ(ξi) exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
pairings
of ξ1,...,ξ2N
pairs
K−1(paired variables)
where paired variables means couples (ξi, ξj) whose difference ξi−ξj (or, what
is the same, ξj − ξi as K−1 is symmetric) is the argument of K−1. The factor
I0 is just the normalization of the vacuum state I0 = 〈0|0〉 which is set to 1.
The inverse kernel K−1 can be found from its definition:
d3x′K(y − x′)K−1(x′ − x) = δ3(x− y) ∀ x,y (54)
leading to:
K−1(x′ − x) = 1
(2π)3
e−ip·(x
′−x) = 〈0|Ψ(x′)Ψ(x)|0〉 (55)
The last equality comes from (53) and (49) in the special case N = 2 or can
be proved directly from field Fourier expansion (42).
We are now in a position to solve the functional integral (47) at finite volume.
First, the functional integration variables are separated from those which are
not integrated, i.e. the field values out of V (V̄ denotes the complementary of
|〈ψ|0〉|2= |N |2 exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
= |N |2 exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
× exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
× exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
where we have taken advantage of the symmetry of the kernel K. The (56)
is a gaussian functional, with a general quadratic form in the field values in
the region V ; it can be integrated in (47) according to standard rules [12],
yielding:
Dψ |〈0|ψ〉|2ψ(x′)ψ(x) = K−1V (x′,x)|N |2 det
]−1/2
exp [
d3x1d
d3x2d
3x′2K
V (x1,x
1)K(x1 − x2)K(x′1 − x′2)ψ(x′2)ψ(x2)
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
= K−1V (x
Dψ |〈0|ψ〉|2 = K−1V (x′,x)〈0|PV |0〉 (57)
The function K−1V is the inverse of K over the region V , namely the inverse
K(x′ − x)ΘV (x′)ΘV (x) ≡ KV (x′ − x) , (58)
the function ΘV (x) being the Heaviside function:
ΘV (x) =
1 if x ∈ V
0 otherwise.
The inverse kernel K−1V fulfills, by definition, the integral equation:
d3x′KV (y − x′)K−1V (x′,x) = δ3(x− y) ∀ x,y ∈ V (60)
Note that, because of the finite region of integration, the inverse kernel may
now depend on both the space variables instead of just their difference. Also
note that K−1V is real and symmetric, being KV real and symmetric. Therefore,
the result of the functional integration yields the simple formula:
〈0|Ψ(x′)PVΨ(x)|0〉 = K−1V (x′,x)〈0|PV |0〉 (61)
where the factor 〈0|PV |0〉 is a positive constant which we will leave unex-
panded.
Altogether, the presence of the projector PV in the eq. (61) modifies the two-
point correlation function by introducing a constant factor 〈0|PV |0〉 and re-
placing the inverse kernel K−1 with a different one K−1V . It can be easily
proved, by using the general formulae of gaussian integrals, that this holds
true in the more general case of many-points correlation function. In fact, the
(53) holds for general quadratic forms in the field ψ and so the eq. (61) can
be generalized to:
Ψ(xn)PV
n=N+1
Ψ(xn)|0〉 = 〈0|PV |0〉
pairings
of x1,...,x2N
pairs
K−1V (paired var.) .
The remaining task is to calculate the inverse kernel K−1V by means of (60).
In fact, we will look for a solution of the more general equation:
d3x′K(y− x′)K−1V (x′,x) = δ3(x− y) ∀ x ∈ V,y (63)
with unbounded y. It is clear that a solution K−1V of equation (63) is also
a solution of (60) because KV equals K when y ∈ V . Setting y unbounded
allows us to find an implicit form for K−1V . In fact (63) implies:
(2π)3
d3x′ eip·x
2ε K−1V (x
′,x) =
eip·x
(2π)3
which is obtained multiplying both sides of (63) by eip·y /(2π)3 and integrating
over the whole space in d3y.
We are now in a position to accomplish our task of calculating 〈p|PV |p〉. By
plugging (61) into (46) we get:
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x′ eip·(x−x
′) 2ε K−1V (x
′,x)〈0|PV |0〉 .
The integration domain in (65) can be split into the region V and the com-
plementary V̄ for both variables. The inverse kernel K−1V is not defined out of
V and can then be set to an arbitrary value, e.g. zero. Otherwise, even if one
chose a non vanishing prolongation of K−1V , an integration outside the domain
V would involve the degrees of freedom of the field out of V and, according to
the general discussion at the end of Sect. 4, the contributing term should be
dropped. Therefore, retaining only the physically meaningful term, the (65)
turns into:
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x′ eip·(x−x
′) 2ε K−1V (x
′,x)〈0|PV |0〉 .
In the above equation one can easily recognize the complex conjugate of the
left hand side of (64). Hence, replacing it with the complex conjugate of the
right hand side, one gets:
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x 〈0|PV |0〉 =
(2π)3
〈0|PV |0〉 (67)
which is the same result of NRQM in (33) in the simple case N = 1, times
a factor 〈0|PV |0〉. This factor still contains a dependence on the field degrees
of freedom out of V , according to the projector expression in eq. (41), which
should disappear at some point. However, we will see that this factor appears
at any multiplicity and therefore becomes irrelevant for the calculations of the
statistical averages.
5.2 Charged scalar field
The calculation done for a neutral scalar field can be easily extended to a
charged scalar field. The 2-component charged scalar field in Schrödinger rep-
resentation reads:
Ψ(x) =
a(p) eip·x +b†(p) e−ip·x
Ψ†(x) =
b(p) eip·x +a†(p) e−ip·x
where a, a† and b, b† are annihilation and creation operators of particles and
antiparticles respectively. They satisfy commutation relations:
[a(p), a†(p′)] = [b(p), b†(p′)] = δ3(p− p′) (69)
[a(p), b(p′)] = [a†(p), b(p′)] = 0
Likewise, the fields obey the commutation relations:
[Ψ(x),Ψ†(y)] = 0 . (70)
and it is then possible to construct field states |ψ, ψ†〉. The projector PV can
be written as:
D(ψ†, ψ)|ψ, ψ†〉〈ψ, ψ†| (71)
with suitable state normalization and arbitrary field functions ψ(x) out of the
region V 4 Similarly to eq. (45), one can write:
4 The functional measure in the equation (71) reads
dψ(x)dψ∗(x)/iπ. Anyhow,
its explicit form is not important for our purposes.
〈0|a(p)= 〈0| 1
d3x e−ip·x
2ε Ψ(x) (72)
a†(p)|0〉= 1
d3x eip·x
2ε Ψ†(x) |0〉
〈0|b(p)= 〈0| 1
d3x e−ip·x
2ε Ψ†(x)
b†(p)|0〉= 1
d3x eip·x
2ε Ψ(x) |0〉
The chain of arguments of the previous subsection can be repeated and the
functional integral:
Ψ(xn)PV
Ψ†(x′n)|0〉 =
D(ψ†, ψ) |〈0|ψ, ψ†〉|2
ψ(xn)ψ
†(x′n)
found to be a multiple gaussian integral. Letting ρ be a permutation of the
integers 1, . . . , N , the integration on the right hand side of eq. (73) yields:
Ψ(xn)PV
Ψ†(x′n)|0〉 = 〈0|PV |0〉
K−1V (xn,x
ρ(n)) (74)
which differs from the corresponding expression for the real scalar field because
now the field is complex and ψ can only be coupled to ψ† [13]:
D(ψ†, ψ)
ψ(ξn)ψ
†(ξ′n)
d3x2 ψ
†(x1)K(x1 − x2)ψ(x2)
K−1(ξn − ξ′ρ(n)) (75)
However, the functional integral involving only two fields ψ and ψ† yields the
same result as for neutral particles. Thus, the kernel K is still the same and
so is the integral equation (64) defining K−1V . The expectation value of PV
in a state with only one particle (or antiparticle) will also be the same as in
eq. (67), that is:
〈0|a(p)PV a†(p)|0〉 = 〈0|b(p)PV b†(p)|0〉 =
(2π)3
〈0|PV |0〉 . (76)
6 Multiparticle channels
We have seen in the previous section that the expectation value 〈p|PV |p〉 for a
single spinless particle is the same obtained in a NRQM approach [5] times an
overall immaterial factor 〈0|PV |0〉. In this section, we will tackle the calcula-
tion of the general multiparticle state. We will see that, by using the projector
definitions in eqs. (41),(71) and subtracting the spurious contributions stem-
ming from external field degrees of freedom, the final result is still the same
as in the NRQM calculation times the factor 〈0|PV |0〉.
We will first address the case of N charged particles.
6.1 Identical charged particles
We will consider a state with N identical charged particles; for N antiparticles
the result is trivially the same.
We want to calculate:
〈N, {p}|PV |N, {p}〉 = 〈0|
a(pn)PV
a†(pn)|0〉 . (77)
Since:
[a†,Ψ†] = [a,Ψ] = [b,Ψ†] = [b†,Ψ] = 0 (78)
one can replace creation and annihilation operators with their expressions in
(72) disregarding the position of the operators with respect to the vacuum
state. In formula:
〈N, {p}|PV |N, {p}〉=
(2π)3
d3x′n e
−ipn·(xn−x′n) 2εn
Ψ(xn)PV
Ψ†(x′n)|0〉
(2π)3
d3x′n e
−ipn·(xn−x′n) 2εn
K−1V (xn,x
ρ(n))〈0|PV |0〉 , (80)
where SN is the permutation group of rank N . Now, like for the single particle
case, we restrict the integration domain to V in (79) in order to get rid of
external degrees of freedom and, by repeatedly using eq. (64), we are left
with:
〈N, {p}|PV |N, {p}〉
〈0|PV |0〉
(2π)3
d3x′n e
−ipn·(x′ρ(n)−x
(2π)3
d3x′n e
ρ−1(n)
−pn)·x′n
Hence, using the definition (25):
〈N, {p}|PV |N, {p}〉 =
FV (pρ(n) − pn)〈0|PV |0〉 (82)
which is exactly the expression (33) obtained in NRQM for N identical bosons,
times the factor 〈0|PV |0〉.
6.2 Identical neutral particles
The case of N identical neutral particles is more complicated because of the
possibility of particle pairs creation. This is reflected in the formalism in the
occurrence of many additional terms in working out the eq. (77). We start
with the calculation for a state with two neutral particles of four-momenta p1
and p2 respectively, generalizing to N particles thereafter. In terms of creation
and annihilation operators:
〈p1, p2|PV |p1, p2〉 = 〈0|a(p1)a(p2) PV a†(p2)a†(p1)|0〉 (83)
which we can rewrite using eq. (45) for a(p1) and a
†(p1) as:
〈p1, p2|PV |p1, p2〉=
(2π)3
d3x′1 e
−ip1·(x1−x1′)
× 2ε1〈0|Ψ(x1)a(p2) PV a†(p2)Ψ(x′1)|0〉 . (84)
In order to use the expression (45) for annihilation and creation a(p2) and
a†(p2) operators we have to get them acting on the vacuum state, hence they
should be moved from their position in eq. (84) outwards. This can be done
by taking advantage of the following commutation rules:
[Ψ(x), a(p)] = −
e−ip·x
[a(p)†,Ψ(x)] = −
eip·x
Using (85) and then plugging (45) in eq. (84) we get:
〈p1, p2|PV |p1, p2〉 =
ε2(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
1)〈0|PV |0〉
− 2ε1
(2π)6
d3x′1
d3x2 e
−ip1·(x1−x1′) e−ip2·(x2−x
1) 〈0|Ψ(x2)PVΨ(x1)|0〉
− 2ε1
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x1−x
2) 〈0|Ψ(x′1)PVΨ(x′2)|0〉
4ε1ε2
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x2−x2
×〈0|Ψ(x2)Ψ(x1)PV Ψ(x′1)Ψ(x′2)|0〉 (86)
where advantage has been taken of the fact that [PV ,Ψ(x)] = 0. The eq. (86)
has four different terms, among which only the last was present in the case
of 2 charged particles in Subsect. 6.1. Instead, the first three terms arise from
contractions of the annihilation and creation operators with field operators on
the same side with respect to PV . By using (62) and replacing all unbounded
integrations with integrations over V to eliminate spurious degrees of freedom
of the field, the eq. (86) yields:
〈p1, p2|PV |p1, p2〉
〈0|PV |0〉
ε2(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
− 2ε1
(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x2−x
1) K−1V (x2,x1)
− 2ε1
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x1−x
2) K−1V (x
4ε1ε2
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x2−x2
K−1V (x1,x
V (x2,x
2) +K
V (x1,x
V (x2,x
+K−1V (x1,x2)K
. (87)
We can now use (64) to integrate away the inverse kernel in eq. (87). For
the second term on the right hand side, we choose to integrate exp(−ip2 ·
V (x2,x1) in x2 and for the third term exp(ip1 · x′1)K−1V (x′1,x′2) in x′1.
At the same time, we perform the integration of the last term in an arbitrary
couple of variables with indices 1 and 2, obtaining:
〈p1, p2|PV |p1, p2〉
〈0|PV |0〉
(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
(2π)6
d3x′1 e
−ip1·(x1−x1′) 2ε1
e−ip2·(x1−x
(2π)6
d3x′2 e
−ip1·(x1−x2′) e−ip2·(x1−x
(2π)6
(2π)6
d3x2 e
−i(p1−p2)·(x1−x2)
(2π)6
d3x′2 e
−ip1·(x1−x2′) e−ip2·(x1−x
2) (88)
Four terms in the above sum cancel out and we are left with:
〈p1, p2|PV |p1, p2〉
〈0|PV |0〉
(2π)6
+ | FV (p1 − p2) |2=
FV (pρ(n) − pn) (89)
This is the same result one would obtain for two identical bosons in a NRQM
framework [5]. The second term on the right hand side accounts for the well
known phenomenon of Bose-Einstein correlation in the emission of identical
boson pairs.
Looking back at the whole derivation, we find that the terms arising from
pairings of field variables on the same side with respect to PV have cancelled
with the terms stemming from commutation of annihilation and creation op-
erators with field operators; the only surviving terms are N ! pairings of field
variables on different sides of PV , just like in the case of charged particles. This
cancellation property holds and so, the formula (82) applies to the case of N
neutral particles as well. A proof based on the form of the thermodynamic
limit V → ∞ is given in Appendix B.
6.3 Particle-antiparticle case
For a state with one particle and one antiparticle, the expectation value of PV
reads:
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉 (90)
implying, in view of (72):
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉
(2π)3
d3x′1 e
−ip1·(x1−x1′) 2ε1〈0|Ψ(x1)b(p2)PV b†(p2)Ψ†(x′1)|0〉 .(91)
Like for neutral particles, the b and b† operators are moved outwards to get
them acting on the vacuum state by using the commutators:
[Ψ(x), b(p)] =
e−ip·x
[b(p)†,Ψ†(x)] =
eip·x
By using the eq. (72) and, like in the previous subsection, restricting the
integration to the region V to eliminate external degrees of freedom, we obtain:
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉
ε2(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
1)〈0|PV |0〉
− 2ε1
(2π)6
d3x′1
d3x2 e
−ip1·(x1−x1′) e−ip2·(x2−x
1) 〈0|Ψ†(x2)PVΨ(x1)|0〉
− 2ε1
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x1−x
2) 〈0|Ψ†(x′1)PVΨ(x′2)|0〉
4ε1ε2
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x2−x2
×〈0|Ψ†(x2)Ψ(x1)PV Ψ†(x′1)Ψ(x′2)|0〉 (93)
whose result is (87) except the (last-1)th term, because now:
〈0|Ψ†(x2)Ψ(x1) PV Ψ†(x′1)Ψ(x′2)|0〉
〈0|PV |0〉
= K−1V (x1,x
V (x2,x
2) +K
V (x1,x2)K
which is a consequence of the general expression (74). Again, four out of five
terms cancel out in eq. (93) and we get:
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉
〈0|PV |0〉
(2π)6
where the missing term with respect to the neutral particle case (89) is the
one involving permutations; this is a natural result because particles and an-
tiparticles are, of course, not identical. Hence, the expectation value of PV on
a particle-antiparticle pair shows a remarkable factorization property, that is:
〈0|a(p1)b(p2)PV b†(p2)a†(p1)|0〉 = 〈0|a(p1)PV a†(p1)|0〉〈0|b(p2)PV b†(p2)|0〉
where the = sign holds provided that the integrations are restricted to the
region V .
Extrapolating to the most general case of N+ particles and N− antiparticles,
it can be argued by using the limit V → ∞ that the factorization of the
microcanoncal state weight holds (see Appendix B) at any multiplicity, i.e.
particles and antiparticles behave like two different species:
〈{N+}, {p+}, {N−}, {p−}|PV |{N+}, {p+}, {N−}, {p−}〉
ρ+∈SN+
FV (pρ+(n+) − pn+)
ρ−∈SN
FV (pρ−(n−) − pn−)〈0|PV |0〉 .(97)
7 Summary and discussion
On the basis of eqs. (19), (21), (16) and (82), which applies to charged as
well as neutral particles, and taking into account that the states |N, {p}〉 have
been chosen to be eigenvectors of four-momentum, we can finally write down
the full expression of the microcanonical partition function of a relativistic
quantum gas of neutral spinless bosons as:
Ω= 〈0|PV |0〉
d3x exp[ix · (pρ(n) − pn)] (98)
with P = (M, 0) and the factor 1/N ! has been introduced in order to avoid
multiple counting when integrating over particle momenta. Similarly, on the
basis of eqs. (20) (22) and (97), the microcanonical partition function of a
relativistic quantum gas of charged spinless bosons can be written as:
Ω= 〈0|PV |0〉
N+,N−=0
N+!N−!
N++N−∏
N++N−∑
ρ+∈SN+
d3x exp[ix · (pρ+(n+) − pn+)]
ρ−∈SN
d3x exp[ix · (pρ−(n−) − pn−)] . (99)
The generalization to a multi-species gas of spinless bosons is then easily
achieved:
Ω= 〈0|PV |0〉
d3pnj
ρj∈ SNj
d3x exp[ix · (pρj(nj) − pnj )] . (100)
where N =
j Nj. The formulae (98), (99) and (100) are our final result.
The finite volume Fourier integrals in the above expressions nicely account
for quantum statistics correlations known as Bose-Einstein and Fermi-Dirac
correlations. We stress once more that for a charged quantum gas, particles
and antiparticles can be handled as belonging to distinct species and they
correspond to different labels j in the multi-species generalization of (100).
The expression of the MPF (98) is the same as obtained in a NRQM calculation
in ref. [5], quoted in this work in (24), times an overall factor 〈0|PV |0〉 which is
immaterial for the calculation of statistical averages in the microcanonical en-
semble. More specifically, the expectation value of PV on a free multi-particle
state (see eqs. (82) and (97)) is the same as in the NRQM calculation (33)
times 〈0|PV |0〉. This result has been achieved enforcing a subtraction prescrip-
tion, namely that all terms depending on the degrees of freedom outside the
system region V must be subtracted “by hand” in all terms at fixed multi-
plicities. The factor 〈0|PV |0〉 is still dependent on those spurious degrees of
freedom, according to the PV definition in eq. (41), but this does not affect
any statistical average because it always cancels out. In the thermodynamic
limit, this factor tends to 1 as PV → I and the large volume limit result known
in literature [4] (i.e. eq. (27)) is recovered.
This result looks surprising in a sense because one would have expected, a pri-
ori that quantum relativistic effects would affect the statistical averages bring-
ing in a dependence on the ratio between the Compton wavelength and the
linear size of the region, getting negligible at small values, i.e. when V → ∞.
This is because the condition (32) from which the MPF expressions (23,24)
ensue, no longer holds in quantum field theory at finite volume and only ap-
plies with good approximation at very large volumes, as discussed in Sect. 4.
However, in the calculation of statistical averages, only the projector PV enters
and this implies the summation over all states at different integrated occupa-
tion numbers |Ñ〉V , according to (31). So, even though the coefficients in the
expansion (34) are different from zero and do depend on the volume, it turns
out that summing all of them at fixed N , one gets the same result as in the
NRQM approximation (32). In formula, by using (31):
〈N, {p}|PV |N{p}〉 =
Ñ,{kV }
|〈N, {p}|Ñ, {kV }〉|2
Ñ,{kV }
|〈0|Ñ, {kV }〉|2
{kV }
|〈N, {p}|N, {kV }〉|2NR (101)
where the index NR means that one should make a non-relativistic quantum
mechanical calculation.
This calculation can be extended to the most general case of the microcanon-
ical ensemble of an ideal relativistic quantum gas by fixing the maximal set
of observables of the Poincaré algebra, i.e. spin, third component and par-
ity besides the energy-momentum four-vector. This will be the subject of a
forthcoming publication.
Acknowledgments
We are grateful to F. Colomo and L. Lusanna for stimulating discussions.
A Bogoliubov relations for a real scalar field
They can be derived by first expressing the localized annihilation operator
as a function of the field Ψ and its conjugated moment Ψ̇. For the localized
problem we have:
Ψ(x) =
akuk(x) e
−iεkt+c.c. (A.1)
where k is a vector of three numbers labelling the modes in the region V , εk
is the associated energy and uk is a complete set of orthogonal wavefunctions
over the region V :
u∗k(x)uk(x
′) = δ3(x− x′)
d3xu∗k(x)uk′(x) = δk,k′ (A.2)
and vanishing out of V . Inverting the (A.1) we have:
d3xu∗k(x) e
Ψ(x) (A.3)
which are valid at any time. We now enforce the mapping (35) and replace
the localized field operators at t = 0, i.e. in the Schrödinger representation,
with those in the full Hilbert space. In other words, we replace in (A.3):
Ψ(x) → 1
(2π)3/2
eip·x ap + c.c.
Ψ̇(x) → 1
(2π)3/2
−i√εp√
eip·x ap + c.c. (A.4)
where εp =
p2 +m2, obtaining:
d3pF (k,p)
εk + εp
ap + F (k,−p)
εk − εp
a†p (A.5)
where:
F (k,p) =
(2π)3/2
d3xu∗k(x) e
ip·x (A.6)
that is the Bogoliubov relations (36). We observe that the localized annihila-
tion operator is a non-trivial combination of annihilation and creation opera-
tors in the whole space. However, the term in (A.5) depending on the creation
operator a†p vanishes in the thermodynamic limit, as expected. This is most
easily shown for a rectangular region, where one has uk(x) = exp[ik · x]/
k is given by the eq. (9) and εk =
k2 +m2. Hence:
F (k,−p) ∝ lim
d3x e−i(p+k)·x ∝ δ3(p+ k) (A.7)
and, consequently εk−εp → 0, which make the second term in (A.5) vanishing.
We can use the eq. (A.5) to work out linear relations between Fock space
states of the localized problem and asymptotic Fock space states. These can
be obtained enforcing the destruction of the localized vacuum state:
ak|0〉V = 0
and writing:
|0〉V = α0|0〉+
d3pα1(p)|p〉+
d3p1d
3p2 α2(p1,p2)|p1,p2〉+ . . . (A.8)
Working out such relations is beyond the scope of this paper.
B Multiparticle state with N particles
In order to prove eq. (82) for neutral particles, the first step is to realize that
a general expansion of:
〈0|a1 . . . aNPV a†N . . . a
1|0〉 (B.1)
(ai is a shorthand for a(pi)) must yield, on the basis of what shown for the
case of 2 particles, a sum of terms like these:
d3x1 . . .
n,m=1
e±ipn·xm f(ε1, . . . , εN)〈0|PV |0〉 (B.2)
where f is a generic function involving a sum of ratios or product of any
number of particle energies and ± stands for either a + or a − sign. That (B.2)
ought to be the final expression can be envisaged on the basis of a repeated
application of the equations (45),(85), (62),(64) in turn. The eq. (B.2) can also
be written as:
d3x1 . . .
d3xN e
pn·x1 . . . ei
pn·xn f(ε1, . . . , εN)〈0|PV |0〉 (B.3)
where
∑′ stands for an algebraic sum with terms having either sign.
If we now take the thermodynamic limit V → ∞ of (B.1), one has PV → I,
thence:
〈0|a1 . . . aNPV a†N . . . a
1|0〉 = 〈0|a1 . . . aNa†N . . . a
δ3(pn − pρ(n)) (B.4)
This tells us that the function f ’s in each term (B.2) must reduce to a triv-
ial factor 1 because they would otherwise survive in the thermodynamic limit,
being a factor depending only on the pn’s. Moreover, since the thermodynamic
limit involves only Dirac’s δ’s with differences of two momenta as argument,
there can be only difference of two momenta as argument of exponential func-
tions in (B.3). Finally, by comparing the eq. (B.3) with the (B.4), we conclude
that the only possible expression at finite V is:
〈0|a1 . . . aNPV a†N . . . a
1|0〉 = 〈0|PV |0〉
(2π)3
d3xn e
−i(pn−pρ(n))·xn
(B.5)
which is precisely (82).
For a state with N+ particles and N− antiparticles, the validity of eq. (97) can
be argued with a similar argument, i.e. by constraining the form of general
terms like (B.2) taking advantage of the limit V → ∞. In the case of particles
and antiparticles, the thermodynamic limit tells us that:
〈0|a1 . . . aN+b1 . . . bN−PV a
. . . a
. . . b
= 〈0|a1 . . . aN+b1 . . . bN−a
. . . a
. . . b
ρ+∈SN+
δ3(pn+ − pρ+(n+))
ρ−∈SN
δ3(pn− − pρ−(n−)) (B.6)
and this determines the form of the expression:
〈0|a1 . . . aN+b1 . . . bN−PV a
. . . a
. . . b
at finite V to be (97). The absence of integrals mixing particles with antipar-
ticles momenta such as: ∫
d3xi e
i(pn+−pn−)·xi
is owing to the absence of such differences as arguments of δ’s in eq. (B.6).
References
[1] A. Chodos et al, Phys. Rev. D 9 (1974) 3471.
[2] V. V. Begun, M. Gazdzicki, M. I. Gorenstein and O. S. Zozulya, Phys. Rev. C
70 (2004) 034901.
[3] V. V. Begun, M. Gazdzicki, M. I. Gorenstein, M. Hauer, V. P. Konchakovski
and B. Lungwitz, arXiv:nucl-th/0611075.
[4] M. Chaichian, R. Hagedorn, M. Hayashi, Nucl. Phys. B 92 (1975) 445.
[5] F. Becattini, L. Ferroni, Eur. Phys. J. C 35 (2004) 243.
[6] F. Becattini, J. Phys. Conf. Ser. 5 175 (2005).
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[8] A. E. Strominger, Annals Phys. 146 (1983) 419
A. Iwazaki, Phys. Lett. B 141 (1984) 342
J. D. Brown, J. W. York, Phys. Rev. D 47, (1993) 1420.
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[13] M. Stone The Physics of Quantum Fields Springer, New York.

  We derive the microcanonical partition function of the ideal relativistic
quantum gas of spinless bosons in a quantum field framework as an expansion
over fixed multiplicities. Our calculation generalizes well known expressions
in literature in that it does not introduce any large volume approximation and
it is valid at any volume. We discuss the issues concerned with the definition
of the microcanonical ensemble for a free quantum field at volumes comparable
with the Compton wavelength and provide a consistent prescription of
calculating the microcanonical partition function, which is finite at finite
volume and yielding the correct thermodynamic limit. Besides an immaterial
overall factor, the obtained expression turns out to be the same as in the
non-relativistic multi-particle approach. This work is introductory to derive
the most general expression of the microcanonical partition function fixing the
maximal set of observables of the Poincare' group.

Introduction
The microcanonical ensemble of the relativistic gas is a subject which has
not received much attention in the past. The reason of scarce interest in this
problem is the peculiarity of physical applications, which have been essentially
confined within statistical approaches to hadron production and the bag model
[1]; these are indeed the only cases where involved volumes and particle num-
bers are so small that microcanonical corrections to average quantities become
relevant. Otherwise, the involved energies or volumes are so large that canoni-
cal and grand-canonical ensembles are appropriate for most practical purposes
(e.g. in relativistic heavy ion collisions). Recently [2], it has been pointed out
that the equivalence, in the thermodynamic limit, between grand-canonical,
canonical and microcanonical ensembles does not apply to fluctuations, more
http://arxiv.org/abs/0704.1967v2
in general to moments of multiplicity distributions of order > 1. Indeed, effects
of the difference between statistical ensembles might be unveiled in studying
multiplicity distributions in relativistic nuclear collisions, [3]. In view of these
phenomenological applications, it would be then desirable to have an in-depth
analysis of the microcanonical ensemble of a relativistic quantum gas.
The main difficulty in tackling this problem stems from the need of imposing a
finite volume. This is necessary to have a correct thermodynamic limit because,
being the energy E finite by construction, also V must be finite if the limit
with E/V fixed is to be taken. Strange as it may seem, a full and rigorous
treatment of the relativistic microcanonical ensemble of an ideal gas at finite
volume is still missing. In all previous works on the subject, at some point, the
large volume approximation is introduced; this is tacitly done, for instance,
considering the single-particle level density as continuous, namely replacing
sums over discrete quantum states with momentum space integrations [4]:
d3p . (1)
In a previous work [5], we have derived an expression of the microcanoni-
cal partition function of an ideal relativistic quantum gas with explicit finite-
volume corrections (see eqs. (23),(24)). However, that expression was obtained
in an essentially multi-particle first-quantization framework, which, as pointed
out in ref. [6] should be expected to become inadequate at very low volumes,
comparable with the Compton wavelength of particles. In this regime, under-
lying quantum field effects should become important and pair creation due to
localization an unavoidable effect [7]. Indeed, there are several studies of the
microcanonical ensemble of a free quantum field [8], but all of them, again, at
some point invoke a large-volume approximation. In the limit of large volumes
one obtains the same expressions of the microcanonical partition function and,
consequently, of statistical averages as in the first-quantization multi-particle
approach followed in ref. [5].
The aim of this work is to derive a general expression of the microcanonical
partition function in a full relativistic quantum field framework, valid for any
finite volume, generalizing the results obtained in [5]. We will do this for the
simplest case of an ideal gas of spinless bosons and postpone the treatment
of particles with spin to a forthcoming publication [9]. We will show that
the expression of the microcanonical partition function obtained in ref. [5]
in a non-relativistic multi-particle approach holds provided that a consistent
prescription of subtracting terms arising from field degrees of freedom outside
the considered volume is introduced.
The paper is organized as follows: in Sects. 2 and 3 we will argue about general
features of the microcanonical ensemble for a relativistic system and discuss
several issues concerning a proper definition of the microcanonical partition
function. In Sect. 4 we will cope with the further issues related to the definition
of a microcanonical ensemble for a quantum field at finite volume. Sects. 5
and 6 include the main body of this work, where the microcanonical partition
function is worked out in a quantum field theoretical framework. In Sect. 7 we
will summarize and discuss the results.
2 On the definition of the microcanonical partition function
It is well known that the fundamental tool to calculate statistical averages in
any ensemble is the partition function. For the microcanonical ensemble one
has to calculate the microcanonical partition function (MPF) which is usually
defined as the number of states with a definite value E of total energy:
states
δ(E − Estate) . (2)
For a quantum system, the MPF is the trace of the operator δ(E − Ĥ): 1
Ω = trδ(E − Ĥ) (3)
with proper normalization of the basis states. For instance, for one non-
relativistic free particle, one has to calculate the trace summing over plane
waves normalized so as to 〈p|p′〉 = δ3(p− p′):
Ω = tr(E − Ĥ) =
E − p
〈p|p〉 = 1
(2π)3
d3p δ
E − p
Thereby, one recovers the well known classical expression implying that the
MPF is the number of phase space cells with size h3 and given energy E. This
number is infinite as the volume is unbounded and it is thus impossible to
calculate a meaningful thermodynamic limit at finite energy density. Hence,
one usually considers a system confined within a finite region by modifying the
hamiltonian Ĥ with the addition of infinite potential walls, i.e. setting Ĥ ′ =
Ĥ+ V̂ where Ĥ is the actual internal hamiltonian and V̂ an external potential
implementing infinite walls. Classically, this leads to a finite Ω, namely:
(2π)3
d3p δ
E − p
1 Throughout this work quantum operators will be distinguished from ordinary
numbers by a symbol “ˆ”.
where V is the volume of the region encompassed by the potential walls. 2
Also the corresponding quantum problem can be easily solved and one has:
Ω = tr(E − Ĥ ′) ≡
E − k
where the sum runs over all wave vectors k which, for a rectangular box
with side Li and periodic boundary conditions, are labelled by three integers
(n1, n2, n3) such that ki = niπ/Li. The difficulty of the quantum expression
(6) with respect to the classical one (5) is that, for a given energy E, a set of
integers fulfilling the constraint imposed by the Dirac’s δ:
in general does not exist. Therefore, the MPF vanishes except for a discrete set
of total energies, for which it is divergent. One has a finite result only for the
integral number of states
0 dE Ω(E), that is the number of states with an
energy less than a given E ′, but this is clearly a stepwise and non-differentiable
function of E ′.
This holds for an ideal gas of any finite number of particles: strictly speaking,
the MPF cannot be defined at finite energy and volume as a continous func-
tion. Only in the thermodynamic limit E → ∞ and V → ∞ an expression like
(6) becomes meaningful because it is then possible to replace the sum over
discrete levels with a phase space integration:
cells
(2π)3
d3p . (7)
Therefore, for a truly finite quantum system, one needs a better definition
of microcanonical partition function. A definition which does not suffer from
previous drawbacks is the following:
Ω = trV δ(E − Ĥ) ≡
〈hV |δ(E − Ĥ)|hV 〉 . (8)
where Ĥ is the internal hamiltonian, without external confining potential, and
the |hV 〉’s form a complete set of normalized localized states, i.e. a complete
set of states for the wavefunctions vanishing out of the region V . It should
be stressed that these states are not a basis of the full Hilbert space because
wavefunctions which do not vanish out of V cannot be expanded onto this
basis; thence the notation trV instead of tr meaning that the trace in eq. (8)
is not a proper one. The difference between (8) and a definition like (3) is
that |hV 〉 are not eigenstates of the hamiltonian Ĥ and the right hand side of
2 We will use the same symbol V to denote both the finite region and its volume.
(8) does not reduce to a discrete sum of δ’s. In fact, this is crucial to have a
continuous function of E, unlike (6).
As an example, let us work out the definition (8) for the single free particle
confined in a rectangular box by infinite potential walls and compare it to (4).
A complete set of states for this problem is:
|k〉 =
exp(ik · x) if x ∈ V k = πnx/Lxπ̂i + ny/Ly ĵπnz/Lzk̂
0 if x /∈ V
Therefore, the MPF definition (8) implies:
〈k|δ(E − Ĥ)|k〉 =
d3p |〈k|p〉|2δ
E − p
where we have inserted a resolution of the identity by using a complete set of
states for the full Hilbert space. The sum in (10) can be calculated and yields:
|〈k|p〉|2 =
V (2π)3
d3x exp[i(k− p) · x]
d3x′ exp[ik · (x− x′)] exp[−ip · (x− x′)]
(2π)3
d3x′ δ3(x− x′)] exp[−ip · (x− x′)] = V
(2π)3
where the completeness relation in V :
exp[ik · (x− x′)] = δ3(x− x′) (12)
has been used. Thus, by using (11), the eq. (10) turns into:
(2π)3
d3p δ
E − p
that is the same expression (5) as in the classical case.
The MPF (13) is now manifestly a continuous function of E and, remarkably,
its thermodynamic limit V → ∞ is the same as the thermodynamic limit of
the “pure” quantum expression (6) (because of (7)). Since the only strict re-
quirement for a well defined MPF is to reproduce the correct thermodynamic
limit, for a gas one can choose a definition like (8) instead of Ω = trδ(E− Ĥ ′)
in (6). We emphasize again that in the passage from (6) to (8) the hamilto-
nian embodying an external confining potential is replaced with the internal
hamiltonian while, at the same time, the localized eigenstates of the former
hamiltonian are used to calculate the trace.
3 The microcanonical partition function of a relativistic system
In special relativity, the microcanonical ensemble must include momentum
conservation beside energy’s to fulfill Lorentz invariance. This means that the
MPF definition (8) should be generalized to [5]:
〈hV |δ4(P − P̂ )|hV 〉 , (14)
P being the four-momentum of the system and P̂ the four-momentum op-
erator. The MPF now being a number of states per four-momentum cell, it
is a Lorentz-invariant quantity. The calculation of the MPF is easiest in the
rest-frame of the system, where P = (M, 0) and the four-volume V u, u being
the four-velocity and V the proper volume of the system, reduces to (V,0),
according to the usual formulation of statistical relativistic thermodynamics
[10,11].
The eq. (14) can be further generalized by enforcing the conservation of not
only energy-momentum but of the maximal set of conserved quantities, i.e.
a maximal set of commuting observables built with the generators of the
Poincaré group. To achieve this, one has to replace δ4(P − P̂ ) in (14) with
a generic projector Pi over an irreducible state of the representation of the
Poincaré group [5,6], i.e:
〈hV |Pi|hV 〉 . (15)
This ensemble is still generally defined as microcanonical ensemble and (15)
microcanonical partition function.
In this work, we will confine to a microcanonical ensemble where only energy-
momentum is fixed, i.e. our projector Pi in eq. (15) will be:
Pi = δ
4(P − P̂ ) (16)
and to an ideal quantum gas, i.e. with P̂ being the free four-momentum op-
erator. In fact, it should be stressed that δ4(P − P̂ ) is not a proper projector,
because P2 = aP where a is a divergent constant. This is owing to the fact
that normalized projectors onto irreducible representations cannot be defined
for non-compact groups, such as the space-time translation group T(4). Nev-
ertheless, we will maintain this naming even for non-idempotent operators,
relaxing mathematical rigor, because it will be favorable to adopt the projec-
tion formalism.
It is worth pointing out that the definition (8) involving only the internal (free)
hamiltonian, is much more fit than (6) for a relativistic generalization. Besides
the advantage of restoring continuity in E, discussed in previous section, this
formulation can be easily extended to the full set of conservation laws with-
out major conceptual difficulty. Conversely, had one tried to generalize (6),
one should have defined a finite region and afterwards sought the observables
commuting with the hamiltonian supplemented with an external confining po-
tential. This would have not been an easy task, and, moreover, a maximal set
of observables commuting with the modified Ĥ would not, in general, define a
Poincaré algebra. This is a well known problem in the static bag model where
the translational invariance is manifestly broken and momentum is thus not
conserved. On the other hand, in the definition (15), we deal with the original
Poincaré algebra of unmodified (free) operators and enforce the localization
through the projector onto confined states.
The eq. (15) can be recast as a full trace by inserting a complete set of states
|f〉 into (15):
〈hV |
|f〉〈f |Pi|hV 〉
〈f |Pi
|hV 〉〈hV |f〉 ≡
〈f |PiPV |f〉 = tr [PiPV ] (17)
where
|hV 〉〈hV | , (18)
is, by definition, the projector onto the Hilbert subspace HV of confined states
(i.e. of wavefunctions vanishing out of V ). The formula (17) is the starting
point to carry out a calculation of the MPF at finite volume. The first thing
to do is to expand (17) as a sum of partition functions at fixed multiplicities,
i.e.:
ΩN (19)
for a single species gas and:
Ω{Nj} (20)
for a multi-species gas, where {Nj} = (N1, . . . , NK) is a set of particle mul-
tiplicities for each species j = 1, . . . , K, defining a channel. ΩN or Ω{Nj} are
obtained by summing over all possible values of kinematical variables with
fixed multiplicities. So, if |f〉 ≡ |N, {p}〉 where {p} labels the set of kinemati-
cal variables of particles in the state |f〉, ΩN reads:
〈N, {p}|PiPV |N, {p}〉 . (21)
Likewise, for a multi-species gas, the microcanonical partition function is ex-
pressed as an expansion over all possible channels:
Ω{Nj} =
〈{Nj}, {p}|PiPV |{Nj}, {p}〉 . (22)
and Ω{Nj} is defined as the microcanonical channel weight.
The microcanonical channel weights (22) have been calculated in ref. [5] with
energy-momentum conservation (i.e. using (16) in a multi-particle, first quan-
tization framework. For a single species ideal spinless gas:
d3p1 . . .d
3pN δ
ρ∈ SN
FV (pρ(n) − pn) (23)
while for a multi-species gas of K spinless bosons:
Ω{Nj} =
d3pnj
ρj∈ SNj
FV (pρj(nj) − pnj ) (24)
being N =
j Nj . In eqs. (23),(24) ρj labels permutation belonging to the
permutation group SNj and FV are Fourier integrals over the region V :
FV (p− p′) ≡
(2π)3
d3x ei(p−p
′)·x (25)
If the volume is large enough so as to allow the approximation:
FV (p− p′) =
(2π)3
d3x ei(p−p
′)·x ≃ δ3(p− p′) (26)
the microcanonical channel weights (24) can be resummed explicitely into the
microcanonical partition function according to (20) and one obtains [5]:
limε→0
(2π)4
∫ +∞−iε
−∞−iε
d3y eiP ·y exp
(2π)3
d3p log(1− e−ip·y)−1
A full analytical calculation of eq. (27) is possible only for the limiting case
of vanishing masses (e.g. microcanonical black body). For the massive case,
four-dimensional integrations cannot be worked out analytically and one has
to resort to numerical computation.
The eq. (27) was implicitely obtained in ref. [4] where the first expression
of the microcanonical partition function of a multi-species ideal relativistic
quantum gas was derived as an expansion (20) over channels, by using the
large-volume approximation (7) from the very beginning. This shows that the
approximation (26) is indeed equivalent to the (7), as also demonstrated in
ref. [5]. Noticeably, the MPF definition eq. (14) without any large volume ap-
proximation involves the appearance of Fourier integrals accounting for Bose-
Einstein and Fermi-Dirac correlations in the quantum gas, which do not show
up in the large-volume approximation enforced in ref. [4]. This approach also
allows to investigate further generalizations when the volume is so small that
relativistic quantum field effects must be taken into account.
4 Microcanonical ensemble and field theory
The calculation of the partition function (14) in a quantum field framework
brings in new difficulties with respect to the first-quantization scheme. This
problem has been approached in literature with a functional approach, in-
spired of the usual grand-canonical thermal field theory [8]. However, these
calculations aim at the limit of large volumes and are therefore insensitive to
the difficulties pertaining to the strict requirement of finite volume discussed
in detail in Sect. 2. As a result, for a free field, the derived expressions are
equivalent to the formula (27).
Instead of starting with a functional integration from the very beginning, we
calculate the microcanonical partition function of a free field by first expanding
it at fixed multiplicities like in eqs. (19),(21) (or channels, for multi-species
gas like in eq. (20)), where |N, {p}〉 are Fock space states with definite particle
multiplicity and kinematical variables {p}. To carry out this calculation, we
first need to find an expression of the microcanonical state weight:
ω ≡ 〈N, {p}|PiPV |N, {p}〉 . (28)
By using (16) and choosing |N, {p}〉 as an eigenstate of the four-momentum
operator with eigenvalue Pf =
i pi, the eq. (28) becomes:
ω = δ4(P − Pf) 〈N, {p}|PV |N, {p}〉 , (29)
To calculate ω and, by a further integration, ΩN we need to know the projector
PV . Since PV is defined as the projector onto the Hilbert subspace of localized
states, it can be easily written down in a multi-particle non-relativistic quan-
tum mechanical (NRQM) framework. As an example, for a non-relativistic
spinless single particle, it reads (see also Sect. 2):
|kV 〉〈kV | (30)
where |kV 〉 is a normalized state of the particle confined in a region V , with a
corresponding wavefunction ψkV (x) vanishing out of V . The symbol kV stands
for a set of three numbers labelling the kinematical modes of the confined
states (e.g. discrete wavevectors, or energy and angular momenta) and the set
|kV 〉 form a complete set of states for the wavefunctions vanishing out of V .
The projector (30) can be easily extended to the many-body case and:
Ñ ,{k}
|Ñ, {kV }〉〈Ñ, {kV }| . (31)
where the symbol {kV } denotes a multiple set of kinematical modes of the
confined states while Ñ is the integrated occupation number, i.e. the sum of
occupation numbers over all single-particle kinematical modes. In the NRQM
approach these numbers are simply particles multiplicities, implying:
〈N, {p}|Ñ, {kV }〉 6= 0 iff N = Ñ . (32)
To calculate the microcanonical state weight, thence the MPF, the products
〈Ñ, {kV }|N, {p}〉 can be worked out on the basis of (32) similarly to what has
been done in Sect. 2 for a single particle, yielding, for a scalar boson [5]:
〈N, {p}|PV |N, {p}〉 =
ρ∈ SN
FV (pρ(n) − pn) (33)
where ρ is a permutation of the integers 1, . . . , N and FV are Fourier inte-
grals (25) over the system region V . From the above equation the expression
of the microcanonical channel weight in eq. (23) follows. We will refer to the
expression (33) as the NRQM one, meaning that is has been obtained in this
first-quantized multi-particle NRQM framework, where, for instance, particles
and antiparticles are simply considered as different species and their contri-
butions factorize.
One could envisage that a projector like (31), written in terms of Fock space
states, could be simply carried over to the relativistic quantum field case,
where |Ñ , {kV }〉 are states of the localized problem, obtained by solving the
free field equations in a box with suitable boundary conditions. Yet, some
difficulties soon arise. First of all, whilst in NRQM the single-particle local-
ized wavefunction |kV 〉 and the free plane wave state |p〉 live in the same
Hilbert space, in quantum field theory the localized and the non-localized
problem are associated with distinct Hilbert spaces. Thus, unlike in NRQM,
it is not clear how to calculate a product like 〈Ñ, {kV }|N, {p}〉. Secondly, even
if there was a definite prescription for it, it should be expected that the inte-
grated occupation numbers of the localized problem do not coincide with actual
particle multiplicities unless the volume is infinite. To understand this point,
one should keep in mind that properly called particles arise from solutions of
the free field equations over the whole space and that the hamiltonian eigen-
states of the localized problem are conceptually different. Consequently, the
integrated particle number operators in the whole space should differ from in-
tegrated number operators within the finite region. Hence, unlike in NRQM,
a state with definite integrated occupation numbers Ñ (we purposely refrain
from calling them particle numbers) should be expected to have non vanish-
ing components on all free states with different numbers of actual particles,
namely:
|Ñ〉V = α0,Ñ |0〉+ α1,Ñ |1〉+ . . .+ αÑ ,Ñ |Ñ〉+ . . . (34)
where α
are non-trivial complex coefficients, and (32) no longer holds. Only
in the large volume limit one expects that the integrated occupation numbers
coincide with actual multiplicities and eq. (32) applies. This kind of effect is
pointed out in the introduction of Landau’s book on quantum field theory
[7]: when trying to localize an electron, an electron-positron pair unavoidably
appears, meaning that the localized single “particle” is indeed a superposition
of many true, asymptotic particle states. Another relevant manifestation of
this difference which is probably more familiar, is the Casimir effect, which is
related to the difference between the true vacuum state |0〉 and the localized
vacuum state |0〉V . Hence, all formulae derived under the approximation (32)
are asymptotic ones, valid in the limit V → ∞ but not at strictly finite volume.
Thus, one should expect significant finite-volume corrections to the eqs. (33)
and the ensuing MPF (23) in a quantum field treatment.
If we want to give an expression like 〈Ñ , {kV }|N, {p}〉 a precise meaning in a
quantum field framework, we first need to map the Hilbert space HV of the
localized problem into the Hilbert space H of the free field over the whole
space. This can be done by mapping the field eigenstates and operators of H
into HV in a natural way as:
Ψ(x)HV −→ Ψ(x)H
|ψ(x)〉HV −→ |ψ(x)〉H (35)
This allows writing linear, non-bijective, Bogoliubov relations expressing the
annihilation and creation operators of the finite region problem as a function
of those of the whole (real scalar) field (see Appendix A for derivation):
d3pF (k,p)
εk + εp
ap + F (k,−p)
εk − εp
a†p (36)
where k are triplets of numbers labelling kinematical modes, just like the
aforementioned kV , εk is the associated energy, εp =
p2 +m2 and:
F (k,p) =
(2π)3
d3xu∗k(x) e
ip·x (37)
uk being a complete set of orthonormal wavefunctions for the finite region.
A remarkable feature of relativistic quantum fields is that, unlike in NRQM,
the localized annihilation operators have non-vanishing components onto the
creation operators in the whole space, as shown by (36). This confirms our ex-
pectation that a localized state with definite integrated occupation numbers is
a non-trivial linear combination of states with different particle multiplicities.
Expectedly, as the volume increases, these components become smaller and in
the infinite volume limit one recovers ak = ap (see Appendix A).
Starting from the Bogoliubov relations (36), it should be possible, in principle,
to calculate the coefficients in eq. (34), thence the microcanonical state weight
(29) by using the expansion (31). In fact, we do not really need to do that.
It is more advantageous, as pointed out in ref. [6], to write the projector PV
in terms of field states rather than occupation numbers of field modes within
the finite region. Indeed, in the general definition in eq. (17):
|hV 〉〈hV | (38)
the states |hV 〉 are a complete set of states of the Hilbert space of the localized
problem HV , where the degrees of freedom are values of the field in each
point of the region V , i.e. {ψ(x)} |x ∈ V . Therefore, the above projector is a
resolution of the identity of the localized problem and can be written as (for
a real scalar field):
Dψ|ψ〉〈ψ| (39)
where |ψ〉 ≡ ⊗x|ψ(x)〉 and Dψ is the functional measure; the index V means
that the functional integration must be performed over the field degrees of
freedom in the region V , that is Dψ = ∏x∈V dψ(x). The normalization of
the states is chosen so as to 〈ψ(x)|ψ′(x)〉 = δ(ψ(x) − ψ′(x)) to ensure the
idempotency of PV . If we now want to give expressions like:
〈{Nj}, {p}|PV |{Nj}, {p}〉 (40)
a clear meaning, we should find a way of completing the tensor product in the
projector (39) with the field states outside the region V such a way the scalar
product can be performed unambiguously.
Unfortunately the answer to this question is not unique and the projector
can be extended to H in infinitely many ways. What is important is that the
result of the calculation is independent of how the projector has been extended.
Thus, at the end of the calculation, one has to check whether spurious terms
appear, possibly divergent, depending explicitely on the chosen state of the
field outside V and these terms must be subtracted away. In general, all terms
depending on the degrees of freedom of the field out of V must be dropped
from the final result.
In this work, we will extend the projector with eigenstates of the field, where
the field function ψ(x) is some arbitrary function outside the region V . Thus,
the projector PV (39) is mapped to:
Dψ|ψ〉〈ψ| |ψ〉 ≡ ⊗x∈V |ψ(x)〉 ⊗x/∈V |ψ(x)〉 (41)
where the index V still implies Dψ = ∏x∈V dψ(x). We will see that, with the
definition (41), spurious terms depending on the degrees of freedom outside
V do arise indeed, but that they can be subtracted “by hand” in a consistent
5 Single particle channel
We will start calculating the expectation value 〈p|PV |p〉 of a single parti-
cle channel in the simple cases of neutral and charged scalar fields. This is
preparatory to the general case of multiparticle states in Sect. 6.
5.1 Neutral scalar field
We consider a gas made of one type of spinless boson described by the free
real scalar field 3 (in Schrödinger representation):
Ψ(x) =
a(p) eip·x+a†(p) e−ip·x
where ε ≡
p2 +m2 is the energy, p is the modulus of the three-momentum
and the normalization has been chosen so as to have the following commutation
rule between annihilation and creation operators:
[a(p), a†(p′)] = δ3(p− p′) . (43)
We start writing the one-particle Fock state |p〉 in terms of creation and an-
nihilation operators acting on the vacuum:
〈p|PV |p〉 = 〈0|a(p) PV a†(p)|0〉 . (44)
3 Henceforth, the capital letter Ψ will denote field operators while for field functions
we will use the small letter ψ.
Since PV is defined, according to (41) as a functional integral of eigenvectors
of the field operator Ψ, it is convenient to express creation and annihilation
operators in terms of the field operators. We shall use following expressions
which are the most appropriate for our task:
〈0|a(p)= 〈0| 1
d3x e−ip·x
2ε Ψ(x)
a†(p)|0〉= 1
d3x eip·x
2ε Ψ(x) |0〉 . (45)
These expressions can be easily checked by plugging, on the right hand side,
the field operators in (42). By using eq. (45) in (44):
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x′ eip·(x−x
′) 2ε 〈0|Ψ(x′)PVΨ(x)|0〉 (46)
It can be easily checked now that the rightmost factor in the above equation
turns out to be (by using the definition (41)):
〈0|Ψ(x′)PVΨ(x)|0〉 =
Dψ |〈ψ|0〉|2ψ(x′)ψ(x) . (47)
where ψ(x) and ψ(x′) are field functions or the eigenvalues of the field operator
relevant to the state |ψ〉, that is:
Ψ(x)|ψ〉 = |ψ〉ψ(x) . (48)
It is possible to find a solution of the functional integral (47) by first consid-
ering the infinite volume limit, when the projector PV reduces to the identity.
In this limiting case, the functional integral in (41) is now performed over
all possible field functions and eq. (41) becomes a resolution of the identity;
〈0|Ψ(x′)Ψ(x)|0〉 is just the two-point correlation function that we write, ac-
cording to (47):
〈0|Ψ(x′)Ψ(x)|0〉 =
Dψ |〈ψ|0〉|2ψ(x′)ψ(x) (49)
The product 〈ψ|0〉 is known as the vacuum functional and reads [12], for a
scalar neutral field:
〈ψ|0〉 = N exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
where N is a field-independent normalization factor, which is irrelevant for
our purposes. The function K(x′ −x) is called kernel and fulfills the equation
[12]: ∫
d3x′ e−ip·x
K(x′ − x) = 2ε e−ip·x (51)
whose solution is:
K(x′ − x) = 1
(2π)3
d3p e−ip·(x
′−x) 2ε . (52)
The functional integral (49) is therefore a gaussian integral and can be solved
by using the known formulae for multiple gaussian integrals of real vari-
ables [12]:
I2N =
ψ(ξi) exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
pairings
of ξ1,...,ξ2N
pairs
K−1(paired variables)
where paired variables means couples (ξi, ξj) whose difference ξi−ξj (or, what
is the same, ξj − ξi as K−1 is symmetric) is the argument of K−1. The factor
I0 is just the normalization of the vacuum state I0 = 〈0|0〉 which is set to 1.
The inverse kernel K−1 can be found from its definition:
d3x′K(y − x′)K−1(x′ − x) = δ3(x− y) ∀ x,y (54)
leading to:
K−1(x′ − x) = 1
(2π)3
e−ip·(x
′−x) = 〈0|Ψ(x′)Ψ(x)|0〉 (55)
The last equality comes from (53) and (49) in the special case N = 2 or can
be proved directly from field Fourier expansion (42).
We are now in a position to solve the functional integral (47) at finite volume.
First, the functional integration variables are separated from those which are
not integrated, i.e. the field values out of V (V̄ denotes the complementary of
|〈ψ|0〉|2= |N |2 exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
= |N |2 exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
× exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
× exp
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
where we have taken advantage of the symmetry of the kernel K. The (56)
is a gaussian functional, with a general quadratic form in the field values in
the region V ; it can be integrated in (47) according to standard rules [12],
yielding:
Dψ |〈0|ψ〉|2ψ(x′)ψ(x) = K−1V (x′,x)|N |2 det
]−1/2
exp [
d3x1d
d3x2d
3x′2K
V (x1,x
1)K(x1 − x2)K(x′1 − x′2)ψ(x′2)ψ(x2)
d3x2 ψ(x1)K(x1 − x2)ψ(x2)
= K−1V (x
Dψ |〈0|ψ〉|2 = K−1V (x′,x)〈0|PV |0〉 (57)
The function K−1V is the inverse of K over the region V , namely the inverse
K(x′ − x)ΘV (x′)ΘV (x) ≡ KV (x′ − x) , (58)
the function ΘV (x) being the Heaviside function:
ΘV (x) =
1 if x ∈ V
0 otherwise.
The inverse kernel K−1V fulfills, by definition, the integral equation:
d3x′KV (y − x′)K−1V (x′,x) = δ3(x− y) ∀ x,y ∈ V (60)
Note that, because of the finite region of integration, the inverse kernel may
now depend on both the space variables instead of just their difference. Also
note that K−1V is real and symmetric, being KV real and symmetric. Therefore,
the result of the functional integration yields the simple formula:
〈0|Ψ(x′)PVΨ(x)|0〉 = K−1V (x′,x)〈0|PV |0〉 (61)
where the factor 〈0|PV |0〉 is a positive constant which we will leave unex-
panded.
Altogether, the presence of the projector PV in the eq. (61) modifies the two-
point correlation function by introducing a constant factor 〈0|PV |0〉 and re-
placing the inverse kernel K−1 with a different one K−1V . It can be easily
proved, by using the general formulae of gaussian integrals, that this holds
true in the more general case of many-points correlation function. In fact, the
(53) holds for general quadratic forms in the field ψ and so the eq. (61) can
be generalized to:
Ψ(xn)PV
n=N+1
Ψ(xn)|0〉 = 〈0|PV |0〉
pairings
of x1,...,x2N
pairs
K−1V (paired var.) .
The remaining task is to calculate the inverse kernel K−1V by means of (60).
In fact, we will look for a solution of the more general equation:
d3x′K(y− x′)K−1V (x′,x) = δ3(x− y) ∀ x ∈ V,y (63)
with unbounded y. It is clear that a solution K−1V of equation (63) is also
a solution of (60) because KV equals K when y ∈ V . Setting y unbounded
allows us to find an implicit form for K−1V . In fact (63) implies:
(2π)3
d3x′ eip·x
2ε K−1V (x
′,x) =
eip·x
(2π)3
which is obtained multiplying both sides of (63) by eip·y /(2π)3 and integrating
over the whole space in d3y.
We are now in a position to accomplish our task of calculating 〈p|PV |p〉. By
plugging (61) into (46) we get:
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x′ eip·(x−x
′) 2ε K−1V (x
′,x)〈0|PV |0〉 .
The integration domain in (65) can be split into the region V and the com-
plementary V̄ for both variables. The inverse kernel K−1V is not defined out of
V and can then be set to an arbitrary value, e.g. zero. Otherwise, even if one
chose a non vanishing prolongation of K−1V , an integration outside the domain
V would involve the degrees of freedom of the field out of V and, according to
the general discussion at the end of Sect. 4, the contributing term should be
dropped. Therefore, retaining only the physically meaningful term, the (65)
turns into:
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x′ eip·(x−x
′) 2ε K−1V (x
′,x)〈0|PV |0〉 .
In the above equation one can easily recognize the complex conjugate of the
left hand side of (64). Hence, replacing it with the complex conjugate of the
right hand side, one gets:
〈0|a(p)PV a†(p)|0〉 =
(2π)3
d3x 〈0|PV |0〉 =
(2π)3
〈0|PV |0〉 (67)
which is the same result of NRQM in (33) in the simple case N = 1, times
a factor 〈0|PV |0〉. This factor still contains a dependence on the field degrees
of freedom out of V , according to the projector expression in eq. (41), which
should disappear at some point. However, we will see that this factor appears
at any multiplicity and therefore becomes irrelevant for the calculations of the
statistical averages.
5.2 Charged scalar field
The calculation done for a neutral scalar field can be easily extended to a
charged scalar field. The 2-component charged scalar field in Schrödinger rep-
resentation reads:
Ψ(x) =
a(p) eip·x +b†(p) e−ip·x
Ψ†(x) =
b(p) eip·x +a†(p) e−ip·x
where a, a† and b, b† are annihilation and creation operators of particles and
antiparticles respectively. They satisfy commutation relations:
[a(p), a†(p′)] = [b(p), b†(p′)] = δ3(p− p′) (69)
[a(p), b(p′)] = [a†(p), b(p′)] = 0
Likewise, the fields obey the commutation relations:
[Ψ(x),Ψ†(y)] = 0 . (70)
and it is then possible to construct field states |ψ, ψ†〉. The projector PV can
be written as:
D(ψ†, ψ)|ψ, ψ†〉〈ψ, ψ†| (71)
with suitable state normalization and arbitrary field functions ψ(x) out of the
region V 4 Similarly to eq. (45), one can write:
4 The functional measure in the equation (71) reads
dψ(x)dψ∗(x)/iπ. Anyhow,
its explicit form is not important for our purposes.
〈0|a(p)= 〈0| 1
d3x e−ip·x
2ε Ψ(x) (72)
a†(p)|0〉= 1
d3x eip·x
2ε Ψ†(x) |0〉
〈0|b(p)= 〈0| 1
d3x e−ip·x
2ε Ψ†(x)
b†(p)|0〉= 1
d3x eip·x
2ε Ψ(x) |0〉
The chain of arguments of the previous subsection can be repeated and the
functional integral:
Ψ(xn)PV
Ψ†(x′n)|0〉 =
D(ψ†, ψ) |〈0|ψ, ψ†〉|2
ψ(xn)ψ
†(x′n)
found to be a multiple gaussian integral. Letting ρ be a permutation of the
integers 1, . . . , N , the integration on the right hand side of eq. (73) yields:
Ψ(xn)PV
Ψ†(x′n)|0〉 = 〈0|PV |0〉
K−1V (xn,x
ρ(n)) (74)
which differs from the corresponding expression for the real scalar field because
now the field is complex and ψ can only be coupled to ψ† [13]:
D(ψ†, ψ)
ψ(ξn)ψ
†(ξ′n)
d3x2 ψ
†(x1)K(x1 − x2)ψ(x2)
K−1(ξn − ξ′ρ(n)) (75)
However, the functional integral involving only two fields ψ and ψ† yields the
same result as for neutral particles. Thus, the kernel K is still the same and
so is the integral equation (64) defining K−1V . The expectation value of PV
in a state with only one particle (or antiparticle) will also be the same as in
eq. (67), that is:
〈0|a(p)PV a†(p)|0〉 = 〈0|b(p)PV b†(p)|0〉 =
(2π)3
〈0|PV |0〉 . (76)
6 Multiparticle channels
We have seen in the previous section that the expectation value 〈p|PV |p〉 for a
single spinless particle is the same obtained in a NRQM approach [5] times an
overall immaterial factor 〈0|PV |0〉. In this section, we will tackle the calcula-
tion of the general multiparticle state. We will see that, by using the projector
definitions in eqs. (41),(71) and subtracting the spurious contributions stem-
ming from external field degrees of freedom, the final result is still the same
as in the NRQM calculation times the factor 〈0|PV |0〉.
We will first address the case of N charged particles.
6.1 Identical charged particles
We will consider a state with N identical charged particles; for N antiparticles
the result is trivially the same.
We want to calculate:
〈N, {p}|PV |N, {p}〉 = 〈0|
a(pn)PV
a†(pn)|0〉 . (77)
Since:
[a†,Ψ†] = [a,Ψ] = [b,Ψ†] = [b†,Ψ] = 0 (78)
one can replace creation and annihilation operators with their expressions in
(72) disregarding the position of the operators with respect to the vacuum
state. In formula:
〈N, {p}|PV |N, {p}〉=
(2π)3
d3x′n e
−ipn·(xn−x′n) 2εn
Ψ(xn)PV
Ψ†(x′n)|0〉
(2π)3
d3x′n e
−ipn·(xn−x′n) 2εn
K−1V (xn,x
ρ(n))〈0|PV |0〉 , (80)
where SN is the permutation group of rank N . Now, like for the single particle
case, we restrict the integration domain to V in (79) in order to get rid of
external degrees of freedom and, by repeatedly using eq. (64), we are left
with:
〈N, {p}|PV |N, {p}〉
〈0|PV |0〉
(2π)3
d3x′n e
−ipn·(x′ρ(n)−x
(2π)3
d3x′n e
ρ−1(n)
−pn)·x′n
Hence, using the definition (25):
〈N, {p}|PV |N, {p}〉 =
FV (pρ(n) − pn)〈0|PV |0〉 (82)
which is exactly the expression (33) obtained in NRQM for N identical bosons,
times the factor 〈0|PV |0〉.
6.2 Identical neutral particles
The case of N identical neutral particles is more complicated because of the
possibility of particle pairs creation. This is reflected in the formalism in the
occurrence of many additional terms in working out the eq. (77). We start
with the calculation for a state with two neutral particles of four-momenta p1
and p2 respectively, generalizing to N particles thereafter. In terms of creation
and annihilation operators:
〈p1, p2|PV |p1, p2〉 = 〈0|a(p1)a(p2) PV a†(p2)a†(p1)|0〉 (83)
which we can rewrite using eq. (45) for a(p1) and a
†(p1) as:
〈p1, p2|PV |p1, p2〉=
(2π)3
d3x′1 e
−ip1·(x1−x1′)
× 2ε1〈0|Ψ(x1)a(p2) PV a†(p2)Ψ(x′1)|0〉 . (84)
In order to use the expression (45) for annihilation and creation a(p2) and
a†(p2) operators we have to get them acting on the vacuum state, hence they
should be moved from their position in eq. (84) outwards. This can be done
by taking advantage of the following commutation rules:
[Ψ(x), a(p)] = −
e−ip·x
[a(p)†,Ψ(x)] = −
eip·x
Using (85) and then plugging (45) in eq. (84) we get:
〈p1, p2|PV |p1, p2〉 =
ε2(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
1)〈0|PV |0〉
− 2ε1
(2π)6
d3x′1
d3x2 e
−ip1·(x1−x1′) e−ip2·(x2−x
1) 〈0|Ψ(x2)PVΨ(x1)|0〉
− 2ε1
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x1−x
2) 〈0|Ψ(x′1)PVΨ(x′2)|0〉
4ε1ε2
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x2−x2
×〈0|Ψ(x2)Ψ(x1)PV Ψ(x′1)Ψ(x′2)|0〉 (86)
where advantage has been taken of the fact that [PV ,Ψ(x)] = 0. The eq. (86)
has four different terms, among which only the last was present in the case
of 2 charged particles in Subsect. 6.1. Instead, the first three terms arise from
contractions of the annihilation and creation operators with field operators on
the same side with respect to PV . By using (62) and replacing all unbounded
integrations with integrations over V to eliminate spurious degrees of freedom
of the field, the eq. (86) yields:
〈p1, p2|PV |p1, p2〉
〈0|PV |0〉
ε2(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
− 2ε1
(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x2−x
1) K−1V (x2,x1)
− 2ε1
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x1−x
2) K−1V (x
4ε1ε2
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x2−x2
K−1V (x1,x
V (x2,x
2) +K
V (x1,x
V (x2,x
+K−1V (x1,x2)K
. (87)
We can now use (64) to integrate away the inverse kernel in eq. (87). For
the second term on the right hand side, we choose to integrate exp(−ip2 ·
V (x2,x1) in x2 and for the third term exp(ip1 · x′1)K−1V (x′1,x′2) in x′1.
At the same time, we perform the integration of the last term in an arbitrary
couple of variables with indices 1 and 2, obtaining:
〈p1, p2|PV |p1, p2〉
〈0|PV |0〉
(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
(2π)6
d3x′1 e
−ip1·(x1−x1′) 2ε1
e−ip2·(x1−x
(2π)6
d3x′2 e
−ip1·(x1−x2′) e−ip2·(x1−x
(2π)6
(2π)6
d3x2 e
−i(p1−p2)·(x1−x2)
(2π)6
d3x′2 e
−ip1·(x1−x2′) e−ip2·(x1−x
2) (88)
Four terms in the above sum cancel out and we are left with:
〈p1, p2|PV |p1, p2〉
〈0|PV |0〉
(2π)6
+ | FV (p1 − p2) |2=
FV (pρ(n) − pn) (89)
This is the same result one would obtain for two identical bosons in a NRQM
framework [5]. The second term on the right hand side accounts for the well
known phenomenon of Bose-Einstein correlation in the emission of identical
boson pairs.
Looking back at the whole derivation, we find that the terms arising from
pairings of field variables on the same side with respect to PV have cancelled
with the terms stemming from commutation of annihilation and creation op-
erators with field operators; the only surviving terms are N ! pairings of field
variables on different sides of PV , just like in the case of charged particles. This
cancellation property holds and so, the formula (82) applies to the case of N
neutral particles as well. A proof based on the form of the thermodynamic
limit V → ∞ is given in Appendix B.
6.3 Particle-antiparticle case
For a state with one particle and one antiparticle, the expectation value of PV
reads:
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉 (90)
implying, in view of (72):
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉
(2π)3
d3x′1 e
−ip1·(x1−x1′) 2ε1〈0|Ψ(x1)b(p2)PV b†(p2)Ψ†(x′1)|0〉 .(91)
Like for neutral particles, the b and b† operators are moved outwards to get
them acting on the vacuum state by using the commutators:
[Ψ(x), b(p)] =
e−ip·x
[b(p)†,Ψ†(x)] =
eip·x
By using the eq. (72) and, like in the previous subsection, restricting the
integration to the region V to eliminate external degrees of freedom, we obtain:
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉
ε2(2π)6
d3x′1 e
−ip1·(x1−x1′) e−ip2·(x1−x
1)〈0|PV |0〉
− 2ε1
(2π)6
d3x′1
d3x2 e
−ip1·(x1−x1′) e−ip2·(x2−x
1) 〈0|Ψ†(x2)PVΨ(x1)|0〉
− 2ε1
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x1−x
2) 〈0|Ψ†(x′1)PVΨ(x′2)|0〉
4ε1ε2
(2π)6
d3x′1
d3x′2 e
−ip1·(x1−x1′) e−ip2·(x2−x2
×〈0|Ψ†(x2)Ψ(x1)PV Ψ†(x′1)Ψ(x′2)|0〉 (93)
whose result is (87) except the (last-1)th term, because now:
〈0|Ψ†(x2)Ψ(x1) PV Ψ†(x′1)Ψ(x′2)|0〉
〈0|PV |0〉
= K−1V (x1,x
V (x2,x
2) +K
V (x1,x2)K
which is a consequence of the general expression (74). Again, four out of five
terms cancel out in eq. (93) and we get:
〈0|a(p1)b(p2) PV b†(p2)a†(p1)|0〉
〈0|PV |0〉
(2π)6
where the missing term with respect to the neutral particle case (89) is the
one involving permutations; this is a natural result because particles and an-
tiparticles are, of course, not identical. Hence, the expectation value of PV on
a particle-antiparticle pair shows a remarkable factorization property, that is:
〈0|a(p1)b(p2)PV b†(p2)a†(p1)|0〉 = 〈0|a(p1)PV a†(p1)|0〉〈0|b(p2)PV b†(p2)|0〉
where the = sign holds provided that the integrations are restricted to the
region V .
Extrapolating to the most general case of N+ particles and N− antiparticles,
it can be argued by using the limit V → ∞ that the factorization of the
microcanoncal state weight holds (see Appendix B) at any multiplicity, i.e.
particles and antiparticles behave like two different species:
〈{N+}, {p+}, {N−}, {p−}|PV |{N+}, {p+}, {N−}, {p−}〉
ρ+∈SN+
FV (pρ+(n+) − pn+)
ρ−∈SN
FV (pρ−(n−) − pn−)〈0|PV |0〉 .(97)
7 Summary and discussion
On the basis of eqs. (19), (21), (16) and (82), which applies to charged as
well as neutral particles, and taking into account that the states |N, {p}〉 have
been chosen to be eigenvectors of four-momentum, we can finally write down
the full expression of the microcanonical partition function of a relativistic
quantum gas of neutral spinless bosons as:
Ω= 〈0|PV |0〉
d3x exp[ix · (pρ(n) − pn)] (98)
with P = (M, 0) and the factor 1/N ! has been introduced in order to avoid
multiple counting when integrating over particle momenta. Similarly, on the
basis of eqs. (20) (22) and (97), the microcanonical partition function of a
relativistic quantum gas of charged spinless bosons can be written as:
Ω= 〈0|PV |0〉
N+,N−=0
N+!N−!
N++N−∏
N++N−∑
ρ+∈SN+
d3x exp[ix · (pρ+(n+) − pn+)]
ρ−∈SN
d3x exp[ix · (pρ−(n−) − pn−)] . (99)
The generalization to a multi-species gas of spinless bosons is then easily
achieved:
Ω= 〈0|PV |0〉
d3pnj
ρj∈ SNj
d3x exp[ix · (pρj(nj) − pnj )] . (100)
where N =
j Nj. The formulae (98), (99) and (100) are our final result.
The finite volume Fourier integrals in the above expressions nicely account
for quantum statistics correlations known as Bose-Einstein and Fermi-Dirac
correlations. We stress once more that for a charged quantum gas, particles
and antiparticles can be handled as belonging to distinct species and they
correspond to different labels j in the multi-species generalization of (100).
The expression of the MPF (98) is the same as obtained in a NRQM calculation
in ref. [5], quoted in this work in (24), times an overall factor 〈0|PV |0〉 which is
immaterial for the calculation of statistical averages in the microcanonical en-
semble. More specifically, the expectation value of PV on a free multi-particle
state (see eqs. (82) and (97)) is the same as in the NRQM calculation (33)
times 〈0|PV |0〉. This result has been achieved enforcing a subtraction prescrip-
tion, namely that all terms depending on the degrees of freedom outside the
system region V must be subtracted “by hand” in all terms at fixed multi-
plicities. The factor 〈0|PV |0〉 is still dependent on those spurious degrees of
freedom, according to the PV definition in eq. (41), but this does not affect
any statistical average because it always cancels out. In the thermodynamic
limit, this factor tends to 1 as PV → I and the large volume limit result known
in literature [4] (i.e. eq. (27)) is recovered.
This result looks surprising in a sense because one would have expected, a pri-
ori that quantum relativistic effects would affect the statistical averages bring-
ing in a dependence on the ratio between the Compton wavelength and the
linear size of the region, getting negligible at small values, i.e. when V → ∞.
This is because the condition (32) from which the MPF expressions (23,24)
ensue, no longer holds in quantum field theory at finite volume and only ap-
plies with good approximation at very large volumes, as discussed in Sect. 4.
However, in the calculation of statistical averages, only the projector PV enters
and this implies the summation over all states at different integrated occupa-
tion numbers |Ñ〉V , according to (31). So, even though the coefficients in the
expansion (34) are different from zero and do depend on the volume, it turns
out that summing all of them at fixed N , one gets the same result as in the
NRQM approximation (32). In formula, by using (31):
〈N, {p}|PV |N{p}〉 =
Ñ,{kV }
|〈N, {p}|Ñ, {kV }〉|2
Ñ,{kV }
|〈0|Ñ, {kV }〉|2
{kV }
|〈N, {p}|N, {kV }〉|2NR (101)
where the index NR means that one should make a non-relativistic quantum
mechanical calculation.
This calculation can be extended to the most general case of the microcanon-
ical ensemble of an ideal relativistic quantum gas by fixing the maximal set
of observables of the Poincaré algebra, i.e. spin, third component and par-
ity besides the energy-momentum four-vector. This will be the subject of a
forthcoming publication.
Acknowledgments
We are grateful to F. Colomo and L. Lusanna for stimulating discussions.
A Bogoliubov relations for a real scalar field
They can be derived by first expressing the localized annihilation operator
as a function of the field Ψ and its conjugated moment Ψ̇. For the localized
problem we have:
Ψ(x) =
akuk(x) e
−iεkt+c.c. (A.1)
where k is a vector of three numbers labelling the modes in the region V , εk
is the associated energy and uk is a complete set of orthogonal wavefunctions
over the region V :
u∗k(x)uk(x
′) = δ3(x− x′)
d3xu∗k(x)uk′(x) = δk,k′ (A.2)
and vanishing out of V . Inverting the (A.1) we have:
d3xu∗k(x) e
Ψ(x) (A.3)
which are valid at any time. We now enforce the mapping (35) and replace
the localized field operators at t = 0, i.e. in the Schrödinger representation,
with those in the full Hilbert space. In other words, we replace in (A.3):
Ψ(x) → 1
(2π)3/2
eip·x ap + c.c.
Ψ̇(x) → 1
(2π)3/2
−i√εp√
eip·x ap + c.c. (A.4)
where εp =
p2 +m2, obtaining:
d3pF (k,p)
εk + εp
ap + F (k,−p)
εk − εp
a†p (A.5)
where:
F (k,p) =
(2π)3/2
d3xu∗k(x) e
ip·x (A.6)
that is the Bogoliubov relations (36). We observe that the localized annihila-
tion operator is a non-trivial combination of annihilation and creation opera-
tors in the whole space. However, the term in (A.5) depending on the creation
operator a†p vanishes in the thermodynamic limit, as expected. This is most
easily shown for a rectangular region, where one has uk(x) = exp[ik · x]/
k is given by the eq. (9) and εk =
k2 +m2. Hence:
F (k,−p) ∝ lim
d3x e−i(p+k)·x ∝ δ3(p+ k) (A.7)
and, consequently εk−εp → 0, which make the second term in (A.5) vanishing.
We can use the eq. (A.5) to work out linear relations between Fock space
states of the localized problem and asymptotic Fock space states. These can
be obtained enforcing the destruction of the localized vacuum state:
ak|0〉V = 0
and writing:
|0〉V = α0|0〉+
d3pα1(p)|p〉+
d3p1d
3p2 α2(p1,p2)|p1,p2〉+ . . . (A.8)
Working out such relations is beyond the scope of this paper.
B Multiparticle state with N particles
In order to prove eq. (82) for neutral particles, the first step is to realize that
a general expansion of:
〈0|a1 . . . aNPV a†N . . . a
1|0〉 (B.1)
(ai is a shorthand for a(pi)) must yield, on the basis of what shown for the
case of 2 particles, a sum of terms like these:
d3x1 . . .
n,m=1
e±ipn·xm f(ε1, . . . , εN)〈0|PV |0〉 (B.2)
where f is a generic function involving a sum of ratios or product of any
number of particle energies and ± stands for either a + or a − sign. That (B.2)
ought to be the final expression can be envisaged on the basis of a repeated
application of the equations (45),(85), (62),(64) in turn. The eq. (B.2) can also
be written as:
d3x1 . . .
d3xN e
pn·x1 . . . ei
pn·xn f(ε1, . . . , εN)〈0|PV |0〉 (B.3)
where
∑′ stands for an algebraic sum with terms having either sign.
If we now take the thermodynamic limit V → ∞ of (B.1), one has PV → I,
thence:
〈0|a1 . . . aNPV a†N . . . a
1|0〉 = 〈0|a1 . . . aNa†N . . . a
δ3(pn − pρ(n)) (B.4)
This tells us that the function f ’s in each term (B.2) must reduce to a triv-
ial factor 1 because they would otherwise survive in the thermodynamic limit,
being a factor depending only on the pn’s. Moreover, since the thermodynamic
limit involves only Dirac’s δ’s with differences of two momenta as argument,
there can be only difference of two momenta as argument of exponential func-
tions in (B.3). Finally, by comparing the eq. (B.3) with the (B.4), we conclude
that the only possible expression at finite V is:
〈0|a1 . . . aNPV a†N . . . a
1|0〉 = 〈0|PV |0〉
(2π)3
d3xn e
−i(pn−pρ(n))·xn
(B.5)
which is precisely (82).
For a state with N+ particles and N− antiparticles, the validity of eq. (97) can
be argued with a similar argument, i.e. by constraining the form of general
terms like (B.2) taking advantage of the limit V → ∞. In the case of particles
and antiparticles, the thermodynamic limit tells us that:
〈0|a1 . . . aN+b1 . . . bN−PV a
. . . a
. . . b
= 〈0|a1 . . . aN+b1 . . . bN−a
. . . a
. . . b
ρ+∈SN+
δ3(pn+ − pρ+(n+))
ρ−∈SN
δ3(pn− − pρ−(n−)) (B.6)
and this determines the form of the expression:
〈0|a1 . . . aN+b1 . . . bN−PV a
. . . a
. . . b
at finite V to be (97). The absence of integrals mixing particles with antipar-
ticles momenta such as: ∫
d3xi e
i(pn+−pn−)·xi
is owing to the absence of such differences as arguments of δ’s in eq. (B.6).
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