be required [6]. According to eqs (2.5) and (2.7) this fixes the constant B if required for a definite system. So the selection of that particular system plus Cond. (3.1) fixes the B of the particular convention. The meaning of Cond. (3.1) is infinite temperature at infinite energy density, which is reasonable enough, although, as we have seen, not obligatory. If the Callen postulates [5] held, then Cond. (3.1) could be required for all the thermodynamical systems. Namely S generally increases slower than linear with E, and this is one of the Callen axioms; then at E->¥ the leading term is universally the "artificial" BE, and then (3.1) puts the universal B to 0 in C°. (Note: henceforth a limes ->¥ is meant in such a way that all the other extensives remain finite.)

However, exceptional model systems are known. E.g. if a system can take only finite internal energy, then the limit in (3.1) does not exist [3]. In all known examples such systems have T<0 states at high energies. The simplest example is the case of fixed magnetic dipoles; a good physical realization is the magnetic excitation of LiF [14].

Such exceptional systems may exist. However in axiomatic thermodynamics one must unambigously define "exceptions". Luckily the known exceptional systems do not obey the Callen axioms [5], and in the same time the limit (3.1) does not exist for them, since in them E has a finite upper bound. So Cond. (3.1) can define the B of C° in spite of the known exceptional systems.

The only problematic system would be such a one, which possesses a term linear in E in the convention when other systems do not. Then this system would have a limiting asymptotic temperature at infinite energy, while the others would not. Then it would not fulfil the Callen axioms. If such an exceptional system exists, then it must definitely be excluded when Cond. (3.1) is required. Note that in that asymptotic domain thermodynamic stability ceases (the specific heat matrix degenerates).

The existence of systems with finite maximal temperature were sometimes discussed, see e.g. Hagedorn's idea for the "melting point of the vacuum". Such a system would prevent the further heating of all the systems in thermal connection with it. If all the systems were to possess a universal limiting temperature, then we are still not in C°, and the universal limiting temperature can be transformed into ¥ by (2.5). If some systems were to possess limiting temperatures but not universally, then all having such can be called special systems, and we can turn to the others when requiring Cond. (3.1).

Quite physical systems are, however, known whose vacuum energy density is a positive constant e_o, best examples are (perturbative) Quantum Chromodynamics and the symmetric phase of Grand Unification. If so, then at infinite dilution p->-e_o [15]. However simultaneously E->e_oV->¥, i.e. the limit needed in (3.2) is impossible. So these systems do not disturb the operativity of Cond. (3.2), because for them eq. (3.2) cannot fix anything.

Cond. (3.2), however, would fix A differently for different systems if such systems existed for which e_o<0. Then p->-e_o>0, the leading term in (2.5) would be (A-e_o)V, and Cond. (3.2) would fix A in an e_o-dependent way.

No such physical or model system is known up to now. However again no physical law is known prohibiting this behaviour. Still, such systems could easily be recognized as exceptional ones. Namely, they would infinitely expand into vacuum or into any other low pressure systems, because of the limiting positive pressure. The boundary surface would propagate with light velocity in interstellar space, a phenomenon never seen by astronomers. (One might contemplate about a "survival of fittest", namely that our vacuum has already expanded into the others, because its eo is the lowest. If this were the reason, no e_o<0 system would turn up.) Again, then, we can leave out exceptional systems when applying Cond. (3.2).

Now we turn to particle numbers and chemical potentials. However in general this goal would exceed the frameworks of one paper, since the systems considered may possess any particular number of particle components, these components can be disjoint or can transmute into each other and they may or may not be conserved. In the next Chapter we briefly discuss these possibilities, although mainly this paper will be devoted to a single conserved component.

Notice that by switching on relativity a new phenomenon called pair creation appears. There are two cases: any particle either has an antiparticle, whose quantum numbers are just opposite, or it is its own antiparticle as well. For the first case we may have the electron e- and the positron e+ as example; for the second case the simplest example is the photon g.

With characteristic quantum numbers the particle will obey a balance law; e.g. electric charge conservation prevents electrons to be created at will. However the particle-antiparticle pair has the same charges as the vacuum, thus the process

(5.3) e⁺+e^- <--> ø

is permitted. But if so, then in equilibrium both numbers of particle and antiparticle are not independent, since no conservation constraint may exist for the pairs. As it turns out,

{N° of ptcle - N° of antiptcle} is the extensive N

{N° of ptcle + N° of antiptcle} is prescribed by other extensives

Indeed, for a relativistic Fermi gas with or without interaction one gets Fermi distributions with corrected energy, with µ for the particle and with -µ for the antiparticle [16].

But then the usual "ideal relativistic Fermi gas" without antiparticles is not an equilibrium distribution and does not belong to equilibrium thermodynamics. Nonequilibrium situations may be very interesting, and even relevant for timescales short compared to characteristic times of reactions; however the problem investigated in this paper is connected to equilibria, and we want to remain restricted to that. For a particle which is its own antiparticle the analogon of Reaction (5.3) reads as

(5.4) ng <--> mg

for the condition (5.2). If S's linear term in N were universal at N=0 then Cond. (5.5) would fix the constant C in (2.5) and then, apart from the trivial factor K, the thermodynamic convention C° would be fixed by Conds. (3.1), (3.2) and (5.5). However, first this universality should be proven.

Remember that in this Chapter we deal with a single particle number. By other words, the present study concentrates on systems having the same set of extensives {V,E,N}. This means that for all systems considered the only particle component is the same kind of particle, which, in addition, is "elementary". In this context looking for universality means that we investigate different systems composed of the same kind of particles. The linear term may be universal for one kind of particles and may not for another.

For simpler formulation let us fix the actual particle to a symbolic R, anything may it be, with an antiparticle. Then

(5.6) N º N(R) - N(R_anti)

Then we can prove some Theorems about the universality of Cond. (5.5). (Note that we cannot prove (5.5) itself: a finite constant on the right hand side is a matter of convention. However for families of systems we can see if they are transformable simultaneously to (5.5).) These Theorems will not be proven with mathematic rigour; however I think if needs be, any Proof could be worked out in the necessary details.

Theorem 1: Cond. (5.5) can be simultaneously fulfilled for all systems composed only of R if this particle is a subject of C symmetry.

Proof: With C symmetry configurations must be of equal probability with antiparticles instead of particles and vice versa. Then S(V,E,N)=S(V,E,-N) with an extremum, therefore, at N=0.

Exceptional situations: The above argumentation may be invalid in the presence of an external field preferring R to R_anti or inversely. In this case, however, the system under investigation is not closed and volume forces are present, which often may lead to problems in thermodynamics.

Comment: C symmetry is naïvely expected for systems, but is violated in weak interactions.

If Theorem 1 cannot help, we can turn to a

Theorem 2: Cond. (5.5) can be simultaneously fulfilled for all macroscopic near-equilibrium systems composed only of R if this particle is a subject of CP symmetry, unless some spatial spontaneous symmetry breaking is present.

Proof: In a macroscopic system without built-in anisotropy or long-range structure the parity is averaged out, because it belongs to the spatial configuration. So P is a symmetry by averaging, therefore microscopic CP symmetry implies macroscopic C symmetry for the system, and then we are back at Theorem 1. For a subsystem this argumentation could be invalidated by the anisotropies via gradients in the neighbourhood. However, first order small gradients cause second order deviations in the thermodynamic potentials. Therefore in the limit of "local equilibrium" or "cellular equilibrium" the relevant quantities do not exhibit the effects of gradients; then again the macroscopic averaging results in a C symmetry.

Exceptional situations: The above argumentation may be invalid in the presence of an external field preferring a direction (see gravity, electromagnetic forces, &c.). In this case, however, the system under investigation is not closed and volume forces are present, which often may lead to problems in thermodynamics. Macroscopic quantum coherence may occur in special systems (ferromagnets, superconductors, &c.); they need individual attention and may be exceptional.

Comment: CP symmetry is violated in hyperweak interaction. The only known example up to now is the kaon. So the present argumentation may be invalid for a system of kaons (and antikaons). If this CP violation is a consequence of external hyperweak fields [17], then we are back at the Comment of Theorem 1. If the CP violation comes from a spontaneous symmetry breaking [18], then the symmetric configuration still may be an extremum although not ground state. Anyway, the hyperweak CP violation deserves further attention, but this will not be made in this paper.

If Theorem 2 cannot help, we can turn to a

Theorem 3: Cond. (5.5) can be simultaneously fulfilled for all macroscopic systems in equilibrium, composed only of R if this particle is a subject of CPT symmetry and if spatial spontaneous symmetry breaking is not present.

Proof: In equilibrium a macroscopic system shows symmetry for T reflection, so then microscopic CPT symmetry implies macroscopic CP symmetry for the system, and then we are back at Theorem 2. For evolution through equilibrium states we can use the familiar quasistationary limit: if the change rate is moderate enough to use equilibrium states at all, then the potentials do not feel the change rate.

Exceptional situations: The above argumentation may be invalid if some physical laws or the geometry are explicitly time-dependent. However, then there are problems with the formulation of thermodynamics anyway [19], and with equilibrium.

Comment: CPT symmetry can be derived from Lorentz symmetry + causality. So its existence is doubtful in a time-dependent Universe. However, up to now no reliable tratment of quantum fields exists in a geometry without a temporal Killing vector, therefore it is difficult to define the symmetries of quantum fields in the early Universe.

In addition we formulate a purely phenomenological and microscopy-free

Theorem 4: Cond. (5.5) can be simultaneously fulfilled for all macroscopic systems composed only of particle R if particle-free states of any of such systems hold chemical equilibrium only with particle-free states of others.

Proof: Note that the physical existence of such a system was not involved; we are free to work with all systems, real, hypothetical or unknown, whose entropy function fulfils the conditions usual for S(V,E,N).

Consider a system with even entropy function S₍₁₎(V,E,N)=S₍₁₎(V,E,-N). For such a system

(5.7) µ₍₁₎(V,E,N=0) = 0

because of the symmetry, i.e. System 1 fulfils Cond. (5.5) in a special convention. Now, take any second system from the set involved in the theorem. According to our assumption and eq. (5.7):

(5.8) µ₍₁₎(E₍₁₎/V₍₁₎,0) = µ₍₂₎(E₍₂₎/V₍₂₎,0) = 0

Hence ¶S/¶N=0 at N=0, that is, in the same convention Cond. (5.5) holds for System 2 too. And so on.

Here we stop. Our conclusion is, then, that in a wide class of systems (of reasonable behaviour) of a single kind of elementary particles there is at least one thermodynamic convention in which Cond. (5.5) simultaneously holds for all of them. Since the possibilities for Conds. (3.1-2) have already been discussed, we can conclude as follows:

All macroscopic systems in equilibrium, with a single kind of particle R (together with its antiparticle R_anti), which obey the Callen axioms and CPT symmetry, can be classified into 2 disjoint classes as follow:

1) those where Conds. (3.1), (3.2) and (5.5) can be simultaneously fulfilled in a convention

C°; and

2) those in which the limit involved in Cond. (3.2) does not exist.

Then we can define the "special" convention C° via the systems of Class 1) (if it is not empty, which we now assume) and then it is uniquely fixed except for K of the temperature unit. Since for the systems of Class 2) the construction is impossible, they do not define C° otherwise.

So far about "elementary" particles. But many systems considered in thermodynamics contains composite entities as particles. And observe that the above Theorems do not help for "composite" particles, e.g. for nonrelativistic macroscopic bodies as billiard balls, (except if they obey CPT symmetry for some reason of their own). This will be shown in details in Sect. 7, here we make only two brief comments.

First about compositeness. We restrain ourselves here from clear-cut definitions. A proton is generally treated as an elementary particle; still by any probability it is composed of 3 quarks. Any particle now believed to be elementary may turn to be composite later. However in the present context we call a particle "elementary" if it may participate in pair creation and annihilation.

This is not a matter of stability. Neutrons are unstable but neutron-antineutron pairs can be produced. In principle anti billiard balls could be created from antiivory, consisting of antiprotons, antineutrons and positrons; and then a billiard ball and an antiball can annihilate. However it is hopeless to pair create billiard balls, since the internal structure could not survive at the energies necessary in the pair creation. So one can expect situations at T ~ Mc² when an "elementary" particle distribution contains more or less equilibrium number of pairs, but cannot expect such for billiard balls. True, some hadrons are at the verge of the "elementary" behaviour in this sense, because the energy needed for creating resonances as excited states is in the same order of magnitude as needed for pair production.

Therefore in equilibrium thermodynamics anti billiard balls or anti protein molecules are practically particle kinds completely independent of billiard balls or protein molecules. This means that C symmetry cannot be expected for the phenomenologic S function, so the Theorems do not help; in the practical situations nothing guarantees that S have an extremum at N=0. Then one can fix the other two zero points, but afterwards

(5.9) lim_n->0 ¶s/¶n = n(e)

Were this n constant, still Transformation (2.5) could bring it to 0, so in the phenomenologic thermodynamics of the given composite "particle" the zero point of µ/T could be fixed according to the scheme suggested. However n(e) is generally not constant; e.g. for Boltzmann gases it takes the form -mc²/T

Of course one may try with a double limit. If V is fixed, N -> 0 and E -> 0 or ¥, then n is a constant which can be removed by the remaining transformation. However -mc²/T diverges at E->0, so the removed constant would be infinite creating ambiguities. The E -> ¥ limit may be free of this problem. This point deserves further elaboration, which however will not be done in the present paper; but a composite body may not survive this limit.

Let us repeat that the zero point fixing is necessary for each kind of particles separately, because each existing, possible or imaginary object which may be a particle in a thermodynamic system is an individual N^I on its own right. Therefore Transformation (2.5) works on S separately according to each N^I; there are as many different C_I-s as the number of possible particle kinds. If one is building up the thermodynamics of billiard balls, then in that formalism the billiard balls are particles on their own right and their chemical potential needs a zero point fixing.

Still one can work in a way similar to that for systems with limiting temperature. There we fixed the zero point of temperature for the "generic" systems, and then in equilibrium of "generic" and "exceptional" systems the temperature scale of the latter ones is fixed too. Now we may fix first the zero points for "elementary" particles, and then in systems containing a composite "particle" and its "elementary" constituents as well the chemical potential of the composite one will be fixed too. There remain problems with composite particles whose composition is not known or which never were in equilibrium with anything else. The zero points of such systems are not fixed operatively by the present condition.

Now, at least for equilibrium systems with a single kind of particle, with CPT symmetry and fulfilling the Callen axioms, we are in the position to compare the conventions C° "simplest in thermodynamic sense" and C' "nearest to mechanics". Let us repeat how these two conventions are defined. In C° the entropy functions of "generic" (non-exceptional) systems obey the simple conditions

lim _E->0 S = 0

(6.1) lim _E->¥ ¶S/¶E = 0

lim _V->¥ ¶S/¶V = 0

lim _N->0 ¶S/¶N = 0

Then from C° a general convention C is reached by the general permitted transformation

(6.2) S -> S +AV +BE + CN

and then any convention C, relative to C°, is defined by the triad of constants:

(6.3) C = C(A,B,C)

Now, C' is the convention in which the thermodynamic pressure

(6.4) p(C') = (¶S(C')/¶V)/(¶S(C')/¶E)

is "as near to" the dynamical pressure P of mechanics "as possible".

In usual "local" mechanics the zero point of P cannot be measured, only its gradients. However, in General Relativity P (i.e. the average spatial diagonal component of the energy-momentum tensor) directly influences the curvature, so, having the curvature measured, P can be determined backwards. Then we can choose an appropriate system, determine its P from observation, and compare it to p, determined by pure thermodynamics. By this way we measure (A,B,C) of C'. Afterwards if one formulates a statement in C°, it can easily and uniquely reformulated in C' and vice versa. Our appropriate system, where P will be measured and p(A',B',C') compared to it, will be the Universe.

We choose the Universe for various reasons. First, it is the only system without exterior, so the only one for which no external disturbing effects are possible. Second, it is the only system of which all other systems, thermodynamical or not, are subsystems, in contact with it. Third, at least in the present Universe, the dynamical pressure P possesses "almost" the same variables as p, which is convenient for comparison.

The first two statements do not need explanation; let us clarify the third. In the presence of transport processes P cannot be equal to p, since the first one includes transport contributions, depending on such non-thermodynamic variables as velocity and temperature derivatives &c. In the simplest near-equilibrium case there are at least 3 such contributions, from shear and bulk viscosities and heat conduction. Symbolically we can write

P = p*(e,n) + (2h(e,n)+3h'(e,n))u^r_;r + q^r_;r

(6.5) qⁱ = -k(e,n)T,_r(g^ir+uⁱu^r)

Here p* may or may not be equal to p(C), but at least has the same variables. The above condition "be as near as possible" can mean only

(6.6) p(C') = p*

Now, in some physical systems the transport contributions can be substantial. However in the Universe shear viscosity and heat transfer are impossible (on large scale) due to the cosmological principle (homogeneity + isotropy), and in the present state the bulk viscosity is negligible [20].

So let us observe the Universe, determine C', and try to measure A', B' and C' belonging to this convention. Here we refer some results of [2].

Up to now there are no evidences that C° and C' would differ at all. The available (scant) bounds come from heavy ion physics and cosmology.

1) Eq. (2.5) together with the Callen axioms [5] leads to

(6.7) B ³ 0

otherwise S would not be monotonous with E. Although one may alter the axioms, now we accept this bound.

2) The cosmological calculations for the primordial nucleosynthesis give essentially good results. So C° gives roughly the cosmologic (C') T at 1 MeV [2]. In heavy ion calculations both hadrochemical and dynamical results are not too bad up to T=150 MeV, and heavy ion physics uses p for P. Therefore probably

(6.8) B < 1/(100 MeV)

3) In the Friedmann equations of cosmology the absolute pressure appears. In the present Universe the matter physically seems to be in first approximation an ideal gas and no effect of the constant A is seen [2]. Therefore roughly

(6.9) -10^-6 cm^-3 < A < +10⁺³ cm^-3

This means that up to now there is no evidence against C°=C'. In Ref. 2 we proposed to set the zero points by the Universe itself, which is the most general of all systems. However, still the C_I's remained free.

There is not absolutely impossible to fix some C_I's in a similar manner. Namely, the change of a C_I alters the reversibility of processes in which N^I changes. However, for a conserved particle the only effect is an additive constant in the entropy, which is irrelevant. So up to now no bound has been found in cosmology for the constant C. C of C' may be 0; direct observations are in principle possible for some lepton and baryon fields, but for quarks only indirect data, reconstructed for the early Universe, can be used. The most which can be said is that there is no evidence against C_I in C'. However, we will return to this problem in Sect. 9 on the grounds of the formal simplicity.

(7.6) lim_N1->0 ¶S/¶N¹ = n₁(E/V,N²/V)

lim_N2->0 ¶S/¶N² = n₂(E/V,N¹/V)

The n's come from interactions, and the interactions come from elementary laws, so they cannot be switched off. However, from C symmetry at least

n₁(V,E,N²=0) = 0

(7.7) n₂(V,E,N¹=0) = 0

in the same way as for a single particle. So the double limit N¹,N²->0 can fix C₁ and C₂. (To see this, cf. eqs. (7.1-5).) Therefore if our assumptions hold but eqs. (7.7) do not, then the only possibility is that we are not in C°. So C° has been found if

lim_E->¥ ¶S/¶E = 0

lim_V->¥ ¶S/¶V = 0

(7.8) lim_N1,N2->0 ¶S/¶N¹ = 0

lim_N1,N2->0 ¶S/¶N² = 0

We ignore the exceptional systems.

Very briefly we discuss transmutations. With only two particle components the simplest possible transmutation law is

N¹ <--> N²

as e.g. for nucleons and D resonances. Then, in equilibrium

(7.9) µ₁ = µ₂

(For more particles a number of such linear relations follow with stochiometric numbers in them.)

Now it is easy to see that eq. (7.10) is compatible with the zero point fixing above. Namely, when both particle numbers are 0, both chemical potentials are 0 too.

Let us fix the zero points of 1/T and p/T in the usual way. We are not yet in Co, because addition of CBBN^BB to S is possible. Evaluate S in the usual way in any definite member of this subset of C's. Then

lim_N^(BB)->0 ¶S/¶N^(BB) º

(8.1) -(1/T)lim_N^(BB)->0 µ_BB = n_BB(E/V)

If this empirically determined n_BB is a constant, then it can be removed by means of Transformation (2.5), and then we have arrived at C°. If n_BB turns out to be e-dependent, then we cannot directly find C°.

We might try with a double limit lim_{E->(0 or ¥)} n_1(BB). By construction that limiting value is a constant. However, as told in Sect. 5, for E->0 n_BB is probably infinite. There would be arguments for the finite limiting value in E->¥; however in that limit the billiard balls decay, and the system cannot be investigated.

Now these equations connect the chemical potential of billiard balls to those of elementary particles. Elementary particles can be investigated at temperatures well above the domain of existence of billiard balls. Therefore there are situations with only elementary particles present, and then the zero points of their chemical potentials can be fixed on the way mentioned in Sect. 5. Then, via eq. (9.4), the zero points of µ{IM} and µ{BB} have been fixed, even if the corresponding composite bodies do not obey C symmetry. At lower temperatures via eq. (8.4) the convention C° can be found even for low energy systems of mere billiard balls.

For composite bodies of unknown composition this process is impossible. However then clearly our information about the bodies is incomplete. Unfortunately for them C° cannot be found in that state of knowledge.

We close the paper with a very superficial discussion of the thermodynamics of the early Universe prior to formation of atoms. We do this to see the connection between C° and C' (C^U) in the particle degrees of freedom.

According to present knowledge, before 10^-36 s, i.e. above 10¹⁵ GeV/particle, the matter seems to have been in the symmetric phase of the Grand Unification Theory. In simplest GUT the group of fundamental interactions is SU(5); the "ultimate" elementary particles are [21]:

i) spin 1/2 fermions, i.e. quarks and leptons;

ii) spin 1 bosons mediating interactions, i.e. X and Y (leptoquarks), gluons, W and Z bosons (including the precursor of photon);

iii) spin 0 bosons of self-interactions (Higgs bosons).

In the symmetric phase all fermions behave similarly, and all vector bosons have the same coupling strengths, while the expectation values of all scalar bosons are 0. The independent particle components are the various fermions.

The SU(5) theory contains only 2 particle conservation laws:

Z = const. (electric charge)

B-L = 3Q-L = const. (baryon N° - lepton N°)

All the particle ratios not affected by these conservations were redistributed repeatedly by transmutations according to equipartitions, and the vector boson expectation values were determined by E/V, Z/V and (B-L)/V.

The Universe is unique, so its Z and B-L are unique values. We do not know of any structure formation in the symmetric phase of GUT, so probably permanent inhomogeneities did not exist; fluctuations were correlated within l~hc/T.

Since Z and B-L are conserved in any spontaneously symmetry-broken states of SU(5) GUT, these values can be measured in the present Universe, even if great caution is needed because of uncertainties and of the danger in extending local data to the whole Universe.

In our neighbourhood the matter seems neutral on large scale; no astronomical observation shows electrically charged stars, galaxies or galaxy clusters. As for the whole Universe one can proceed as follows. The so called "Copernican principle" is often meant as suggesting isotropy + homogeneity. Isotropy can be checked on the 2.7 K blackbody radiation, which is indeed isotropic in temperature up to 5 digits. The 10^-5 changes can be explained by e.g. spontaneous thermal fluctuations during primordial inflation at the breakup of the GUT symmetry [22]. As for homogeneity galaxy counts definitely show grouping or "walls" separated by cca. 140 Mpc [23]. So there is no homogeneity on this scale. However this structure seems roughly periodic [24], in which case on larger scales there is no evidence against homogeneity.

Then let us accept maximal spatial symmetry (6 symmetries), as generally done in theoretical cosmology. The only possibilities for the metric tensor with so many symmetries are [25]

ds² = dt² - R²(t){dx²+f²(x)dW²}

sin x (k=+1; SO(4))

(9.1) f(x) = x (k=0; E(3))

sh x (k=-1; SO(3,1))

Now, a noncompensated electric charge would result in a global electromagnetic potential Ai. But the only vector potential with one of the above symmeties is

(9.2) Aⁱ(x^k) = (A^o(t),0)

which form leads to vanishing field tensor. Therefore a homogeneous isotropic Universe would not be possible with homogeneous noncompensated electric charge. Therefore it is better to accept Z=0.

For B-L we can use particle counts, although for some particles it is practically impossible and the data are rather local. In our neighbourhood the overwhelming majority of fermions are protons, neutrons, electrons, neutrinos and antineutrinos. The counts show

n(p) = n(e) (electric neutrality)

n(n) » 0.16*n(p) (from stellar He)

and neutrino counts are still impossible. However

B/L » (n(p)+n(n))/(n(e)+n(n)-n(n_anti))

is in the order of unity if (n(n)-n(n_anti))/n(e) is in the order of unity, which is not impossible. So present data do not rule out

B-L = 0.

In this case all the conserved quantum numbers of the very early Universe would be those of the vacuum.

Then during the symmetric phase of GUT the thermodynamic description needed an entropy function

(9.3) S = Vs(e=E/V)

with a form similar to that of the radiation fields. In addition, near to Planck time and/or energy a further extensive may have been necessary for the cosmologic Hawking radiation [19].

The spontaneous breakdown of GUT symmetry is extensively discussed in the literature. Here it is sufficient to mention that the endproduct is the "standard theory", SU(3)xSU(2)xU(1). In this theory baryon and lepton numbers are independently conserved. In the simplest case this would mean

(9.4) S=Vs(E/V,B/V,L/V)

(According to experiments, the lepton conservation holds in 3 different families separately, which fact now will be ignored for simplicity.) Since baryons and leptons interact, we are back at the situation of Sect. 7. (CP is violated in the SU(2) sector, but CPT may be exact or a good approximation in a sufficiently old Universe.) Now for particle numbers CU is defined so that formulae be simplest. Note that interactions in the B gas include QCD, while in the L gas or between the two ones do not, so they are weaker. Since QCD has asymptotic freedom, in first approximation even in C^U (C') one gets

(9.5) µ_B(V,E,B=0,L) = 0

µ_L(V,E,B,L=0) = 0

Up to now no evidence is known in this era for differences between C° and C^U, i.e. for nonzero C_B and C_L; and for any case they are bound from above by the strengths of non-SU(3) interactions.

However this simple picture becomes rather complicated after the quark confinement transition (rougly between 7 and 12 µs, at 160 MeV temperature [26]). After hadronisation the SU(3) sector is represented by a great diversity of strongly interacting hadrons. Then the interactions may introduce a number of C_I's between C° of eq. (7.8) and C^U of maximal simplicity of formulae. Still, all these interactions go back to 3 fundamental ones, so the C_I's should be expressable with 3 independent coefficients, but the description of hadronic interactions is not yet in this stage.

At the end of the plasma era (t~300000 y, T~3000 K) atoms are formed, then molecules and so on. Thenceforth the world contains very complicated entities as e.g. billiard balls, for which only phenomenologic descriptions are available. Then we are inevitably back to the problems of Sect. 8, but here we stop.