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A constructive scheme for studying systems with spontaneous symmetry breakdown, based on the fundamental notion of quasi-averages (N.N. Bogolyubov [[#References|[1]]], 1961).

A quasi-average is a thermodynamic (in statistical mechanics) or vacuum (in quantum field theory) average of dynamical quantities in a specially modified averaging procedure, enabling one to take into account the effects of the influence of state degeneracy of the system.

In statistical mechanics, under spontaneous symmetry breakdown one can, by using the method of quasi-averages, describe macroscopic observable within the framework of the microscopic approach (cf. also [[Statistical mechanics, mathematical problems in|Statistical mechanics, mathematical problems in]]).

In problems with degeneracy there corresponds to a single energy level more than one independent state of the system; the average  $  \langle  A \rangle $
of any dynamical quantity  $  A $
is defined uniquely:

$$ \tag{1 }
\langle  A \rangle _ {H}  = \ 

\frac{ \mathop{\rm Tr} ( Ae ^ {- \beta H } ) }{ \mathop{\rm Tr}  e ^ {- \beta H } }
 ,
$$

where  $  H $
is the Hamiltonian of the system and  $  \beta $
is the reciprocal of the temperature. If the statistical equilibrium state of the system possesses lower symmetry than the Hamiltonian of the system (so-called spontaneous symmetry breakdown, see [[#References|[2]]]–[[#References|[5]]]), then it is necessary to supplement the averaging operation (1) by a rule forbidding "superfluous" averaging over the values of macroscopic quantities considered for which a change is not accompanied by a change in energy.

This is achieved by introducing quasi-averages, that is, averages over the Hamiltonian supplemented by infinitesimally-small terms that violate the additive conservations laws. Thermodynamic averaging may turn out to be unstable with respect to such a change of the original Hamiltonian, which is another indication of degeneracy of the equilibrium state.

Thus, the quasi-average  $  \prec A \succ $
of a dynamical quantity  $  A $
for the system with Hamiltonian  $  H $
is defined as the limit

$$ \tag{2 }
\prec A \succ _ {H}  = \ 
\lim\limits _ {\nu \rightarrow 0 }  \lim\limits _ {V \rightarrow \infty } \ 
\langle  A \rangle _ {H _  \nu   }  ^ {(} V) ,
$$

where  $  \langle   \cdot  \rangle _ {H _  \nu   } $
denotes the ordinary average taken over the Hamiltonian  $  H _  \nu  $, 
containing the small symmetry-breaking terms introduced by the inclusion parameter  $  \nu $, 
which vanish as  $  \nu \rightarrow 0 $. 
According to definition (2), the ordinary thermodynamic average is obtained by extra averaging of the quasi-average over the symmetry-breaking group.

The value of the quasi-average (2) may depend on the concrete structure of the additional term  $  \Delta H = H _  \nu  - H $, 
if the dynamical quantity  $  A $
to be averaged is not invariant with respect to the symmetry group of the original Hamiltonian  $  H $. 
For a degenerate state the limit of ordinary averages (2) as the inclusion parameters  $  \nu $
of the sources tend to zero in an arbitrary fashion, does not exist. For a complete definition of quasi-averages it is necessary to indicate the manner in which these parameters tend to zero in order to ensure convergence (see, for example, [[#References|[5]]]). On the other hand, in order to remove degeneracy it suffices, in the construction of  $  H $, 
to violate only those additive conservation laws whose "inclusion" leads to instability of the ordinary average. Here, for quasi-averages the selection rules of the correlation functions that are not obeyed are those that are restrained by the above conservation laws.

The method of quasi-averages is directly related to the principle of weak correlation (see [[#References|[8]]], [[#References|[9]]]): The correlation functions

$$ \tag{3 }
\prec U _ {1} ( x _ {1} , t _ {1} ) \dots U _ {n} ( x _ {n} , t _ {n} ) \succ ,
$$

where  $  U _ {s} ( x _ {s} , t _ {s} ) $
are the field functions in the [[Heisenberg representation|Heisenberg representation]]  $  \Psi ( x _ {s} , t _ {s} ) $
or  $  \Psi  ^ {+} ( x _ {s} , t _ {s} ) $, 
split up into products

$$ \tag{4 }
\prec U _ {1} ( x _ {1} , t _ {1} ) \dots U _ {s-} 1
( x _ {s-} 1 , t _ {s-} 1 ) \succ \times
$$

$$ 
\times 
\prec U _ {s} ( x _ {s} , t _ {s} ) \dots U _ {n} ( x _ {n} , t _ {n} ) \succ ,
$$

if the collection of points  $  x _ {1} \dots x _ {s-} 1 $
are infinitely far removed from the collection of points  $  x _ {s} \dots x _ {n} $
for fixed time variables  $  t _ {1} \dots t _ {n} $. 
In the case of state degeneracy under consideration the expressions  $  \prec  \cdot  \succ $
in this formula must necessarily be understood as quasi-averages: The above-mentioned statement of the principle of weak correlation is definitely false if the  $  \prec  \cdot  \succ $
are interpreted as ordinary averages.

For the construction of the non-equilibrium statistical operator one considers infinitesimal perturbations that break the symmetry (of the [[Liouville-equation(2)|Liouville equation]]) with respect to time reversal. The application of this operation is equivalent to choosing retarded solutions of the Liouville equation (see [[#References|[7]]]).

Questions of the choice of method of convergence to zero of the inclusion parameters of the sources  $  \nu $
guaranteeing the convergence of the ordinary averages in the definition of quasi-averages (2) have been considered in the framework of the method of approximating Hamiltonians. Here, the original Hamiltonian is replaced, using a special method of replacing the dynamical quantities that in the limit of the large system commute with the entire algebra of local observables, by  $  c $-
numbers (the so-called approximating Hamiltonian). This last Hamiltonian should be much simpler than the original one (and in a wide range of physically important cases it has an exact solution) and should be equivalent to it as  $  V \rightarrow \infty $(
see [[#References|[7]]]). The additional term  $  \Delta H $
has to be taken proportional to the solution in the general case of the minimax problem for the limit function  $  ( V \rightarrow \infty ) $
of the free energy of the approximating Hamiltonian. Then an arbitrary sequence of real positive  $  \nu $
converging to zero ensures the convergence in definition (2); furthermore, the quasi-averages thus constructed turn out to be equal to the corresponding solutions of this minimax problem.

Within the framework of the method of approximating Hamiltonians an alternative method has been given for defining quasi-averages, without the introduction of extra terms in  $  H $. 
In this approach, in calculating the quasi-average  $  \prec A \succ _ {H} $
the dynamical quantity  $  A $
under consideration is multiplied by a certain factor  $  L $
that converges, in a certain sense, to one as  $  V \rightarrow \infty $, 
and the Hamiltonian  $  H $
remains unchanged [[#References|[4]]]:

$$ \tag{5 }
\prec A \succ _ {H}  =  \lim\limits  \langle  A \cdot L \rangle .
$$

The quasi-averages defined according to (2) and (5) are the same.

The mathematical apparatus of the method of quasi-averages includes the [[Bogolyubov theorem|Bogolyubov theorem]] on singularities of type  $  1 / q  ^ {2} $
and the [[Bogolyubov inequality|Bogolyubov inequality]] for Green and correlation functions (cf. [[Green function|Green function]]; [[Correlation function in statistical mechanics|Correlation function in statistical mechanics]]). It includes algorithms for establishing non-trivial estimates for equilibrium quasi-averages, enabling one to study the problem of ordering in statistical systems and to elucidate the structure of the energy spectrum of the underlying excited states (see [[#References|[4]]], [[#References|[8]]]).

The concept of quasi-averages is indirectly related to the theory of phase transition (see [[#References|[6]]], [[#References|[7]]], [[#References|[9]]]); the instability of thermodynamic averages with respect to perturbations of the Hamiltonian by a violation of the invariance with respect to a certain group of transformations means that in the system transition to an extremal state occurs.

In [[Quantum field theory|quantum field theory]], for a number of model systems it has been proved that there is a phase transition, and the validity of the Bogolyubov theorem on singularities of type  $  1 / q  ^ {2} $
has been established; also, the possibility has been studied of local instability of the vacuum and the appearance of a changed structure in it.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.N. Bogolyubov, "Selected works" , '''3''' , Kiev (1971) (In Russian) {{MR|0414297}} {{ZBL|0221.01014}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> , ''Statistical physics and quantum field theory'' , Moscow (1973) (In Russian) {{MR|}} {{ZBL|1195.81005}} {{ZBL|1153.81302}} {{ZBL|1153.81301}} {{ZBL|1119.81300}} {{ZBL|1110.00014}} {{ZBL|1112.81301}} {{ZBL|1099.81500}} {{ZBL|1062.81500}} {{ZBL|1064.81500}} {{ZBL|1014.00503}} {{ZBL|1062.81501}} {{ZBL|0994.81077}} {{ZBL|1016.81042}} {{ZBL|0967.00039}} {{ZBL|1087.82505}} {{ZBL|1074.81537}} {{ZBL|1014.00509}} {{ZBL|0947.00014}} {{ZBL|0967.00041}} {{ZBL|0956.81504}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.A. Grib, E.V. Damaskinskii, V.M. Maksimov, ''Sov. Phys. Uspekhii'' , '''13''' (1971) pp. 798- ''Uspekhi Fiz. Nauk'' , '''102''' (1970) pp. 587–620</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N.N. Bogolyubov jr., B.I. Sadovnikov, "Some questions in statistical mechanics" , Moscow (1975) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N.N. Bogolyubov jr., "A method for studying model Hamiltonians" , Pergamon (1972) (Translated from Russian) {{MR|0406286}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Brout, "Phase transitions" , New York (1965)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974) {{MR|0289084}} {{ZBL|0997.82506}} {{ZBL|1016.82500}} {{ZBL|0177.57301}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Akhiezer, S.V. Peletminskii, "Methods of statistical physics" , Oxford Univ. Press (1981) (Translated from Russian) {{MR|0635925}} {{ZBL|}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> C.J. Preston, "Gibbs states on countable sets" , Cambridge Univ. Press (1974) {{MR|0474556}} {{MR|0474557}} {{ZBL|0297.60054}} </TD></TR></table>