Particle states

The state F) is such that the particle states a, b, c,..., q are occupied and each particle is equally likely to be in any one of the particle states. However, if two of the particle states a, b, c,...,q are the same then F) vanishes it does not correspond to an allowed state of the assembly. This is a characteristic of antisynmietric states and it is called the Pauli exclusion principle no two identical fennions can be in the same particle state. The general fimction for an assembly of bosons is... 

Sj Uj, and if the yth single-particle state has energy then the energy of the system in the state v is =... 

Wlren the single-particle states j are densely packed within any energy interval of k T, the sum over j can be replaced by an integral over energy such that... 

A2.2.146). The average density N )/V= is the thennodynamic density. At low and high T one expects many more accessible smgle-particle states than the available particles, and (A) = means that each n.p... 

This is the classical Boltzmaim distribution m which (Uj)/(A, tire probability of finding a particle in the single-particle state j, is proportional to the classical Boltzmaim factor... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

We further make the following tentative conjecture (probably valid only under restricted circumstances, e.g., minimal coupling between degrees of freedom) In quantum field theories, too, the YM residual fields, A and F, arise because the particle states are truncated (e.g., the proton-neutron multiplet is an isotopic doublet, without consideration of excited states). Then, it is within the truncated set that the residual fields reinstate the neglected part of the interaction. If all states were considered, then eigenstates of the form shown in Eq. (90) would be exact and there would be no need for the residual interaction negotiated by A and F. 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

Symmetrizationof N-Particle States.—Let Pbe an arbitrary permutation13 among the eigenvalues occurring in the set A. By this we mean the following start with some arbitrary set of vectors A1>, A2>,- , AW>, one for each particle in its respective Hilbert space St then carry out an arbitrary permutation of the values... 

Exactly the same set of eigenvalues appears in this as before, but in a different order among the particles. There are Nl permutations, and if we add all the 2V vectors like that in Eq. (8-92) we shall have a vector that is completely symmetrical in all the particles.14 Thus we may define the symmetrical -particle state as... 

In general whether we are discussing symmetrical or antisymmetrical states, the numbers nx are usually termed the occupation numbers, and if we are given a complete spectrum of one-particle states, indicated by A, then the set of occupation numbers assigned to the values of A, specifies the state of the system of JY identical particles just as well as the assignment of a A-value to each particle. Thus we may use the notation just as well as A to specify the state, and so write... 

The one-particle state of momentum k will be denoted by k>. Since the particle is noninteracting and free, its energy is k0 = Vk2 + ma. We shall normalize the one-particle states in such a way that... 

The normalization of the one-particle states together with the definition of the vacuum now implies that... 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

A state of m particles and n antiparticles can be constructed from the no-particle state 0>, which now is annihilated by both the b and the On operators ... 

The normalization requirement of the one and many particle states is satisfied if we impose the following commutation rules on the b and d... 

In concluding this section we briefly establish the connection between the Dirac theory for a single isolated free particle described in the previous section and the present formalism. If T> is the state vector describing a one-particle state, iV T> = 1 T> consider the amplitude... 

A particular representation of the commutation rules (9-636) and (9-637) characterized by the existence of a no-particle state 0> is exhibited by the equations... 

Let us in fact consider the expectation value of the current operator in the no-particle state in<0 (a ) 0>ln. In order to obtain an insight into this quantity, we first treat the case of a very weak external field so that only effects to first order in the external field need be... 

The representation of these commutation rules is fixed by the requirement that there exist no-particle states 0)in and 0>out which are annihilated by the corresponding positive frequency operators, i.e. 

We further assume the existence of a no-particle state such that... 

Next, let us check whether the theory as formulated above satisfies certain reasonable physical requirements. One of these is dearly the steadiness of the vacuum and one-particle states. The steadiness of the vacuum would correspond to the statement that out<0 0>jn = 1. Let us, therefore, compute this quantity explidtly ... 

In other words that a negaton initially in a state of momentum p, energy Vp2 + m2 helicity s, would remain forever in that state (since it does not interact with anything). Let us, however, compute the left-hand side of Eq. (11-123) with the -matrix given in terms of the interaction hamiltonian (11-121). To lowest order the diagrams indicated in Fig. 11-6 contribute and give rise to the following contribution to the matrix element of S between one-particle states... 

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