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For a finer analysis of this context we refer to our Sect. 5. We shall need to study the chemical potential, as we are going to explain. Chemical potential. Relativistic case. The chemical potential is a label that each charge sets on different equilibrium states at the same temperature. Its structure in Quantum Statistical Mechanics has been explained in the work of Araki, Haag, Kastler and Takesaki [1], see also [2]. The labels appear by considering the extensions of these states from the observable algebra to KMS states of the field algebra (with the time evolution modified by one-parameter subgroups of the gauge group).

We will consider also different temperature states. For a finer analysis of this context we refer to our Sect. 5. We shall need to study the chemical potential, as we are going to explain. Chemical potential. Relativistic case. The chemical potential is a label that each charge sets on different equilibrium states at the same temperature. Its structure in Quantum Statistical Mechanics has been explained in the work of Araki, Haag, Kastler and Takesaki [1], see also [2]. The labels appear by considering the extensions of these states from the observable algebra to KMS states of the field algebra (with the time evolution modified by one-parameter subgroups of the gauge group).

Kallies, A. Özlük, M. Peter, C. Snyder and in case II S(x) = −κ3 x 2 + O(x ρ+6ε ). To compute the constants the Dirichlet series ∞ D(s) := n,v=1 c(v, n) , n2 v s Re(s) > 1, may be used. Then D(3) = 16κ1 , D (3) = 8κ1 − 16κ2 . Using the explicit formula for c(v, n) we get for v ∈ N, ∞ n=1 c(v, n) =A n2 2 1+ 22 p3 2 , −p if 2 v, if 2||v, if 4||v, if 8|v. 40 3 16 A=  32    224 3 From this it follows that 1− D(s) = p>2 2 p3 − p Dp (s), p where 1 40 16 32 224 + s + 2s + , 3 2 2 3 22s (2s − 1) 1 2(p 3 − 1) Dp (s) = 1 + s , p > 2.

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Communications in Mathematical Physics - Volume 222 by M. Aizenman (Chief Editor)


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