Berezin integral

Berezin integral

In mathematical physics, a Berezin integral is a way to define integration for functions of Grassmann variables. It is not an integral in the Lebesgue sense; it is called integration for some analogue properties and since it is used in physics in a parallel manner to ordinary integration. The technique was invented by the physicist David John Candlin in 1956 [cite journal|journal= Nuovo Cimento | author= D.J. Candlin | volume =4 | title = On Sums over Trajactories for Systems With Fermi Statistics|year=1956|pages=224|doi= 10.1007/BF02745446] , but it is named after the Russian mathematician Felix Berezin, who included a treatment in his textbook [A. Berezin, "The Method of Second Quantization", Academic Press, (1966)] .

Definition

The "Berezin integral" is defined to be a linear functional int af( heta)+bg( heta) d heta = aint f( heta)d heta +bint g( heta)d heta

fulfilling the partial integration ruleint fracpartial{partial heta}f( heta)d heta = 0 ;.

These properties define the integral uniquely up to a multiplicative constant which we can set to 1 and translate into the rule

*int (a heta+b) d heta = a ;.

This is the most general function, because every homogeneous function of one Grassmann variable is either constant or linear.

Multiple Variables

Integration over multiple variables is defined by Fubini's theorem:int f_1( heta_1)ldots f_n( heta_n) d heta_1ldots d heta_n = int f_1( heta_1)d heta_1ldots int f_n( heta_n) d heta_n ;.

Note that the sign of the result depends on the order of integration.

Suppose now we want to do a substitution: heta_i= heta_i(xi_j)where as usual (ξj) implies dependence on all ξj. Moreover the function θi has to be an odd function, i.e. contains an odd number of ξj in each summand. The Jacobian is the usual matrixJ_{ij}=frac{partial heta_i}{partialxi_j} ;.

the substitution formula now reads asint f( heta_i)d heta = int f( heta_i(xi_j)) operatorname{det}(J_{ij})^{-1} dxi ;.

ubstitution formula

Consider now a mixture of even and odd variables, i.e. xa and θi. Again we assume a coordinate transformation as x_a=x_a(y_b, heta_j),,; xi_i=xi_i(y_b, heta_j);, where xa are even functions and θi are odd functions. We assume the functions xa and θi to be defined on an open set U in Rm. The functions xa map onto the open set U' in Rm.

The change of the integral will depend on the Jacobian J_{alphaeta}=frac{partial (x_a, heta_i)}{partial(y_b,xi_j)}. This matrix consists of 4 blocks J=egin{bmatrix}A&B\ C&D end{bmatrix}. A and D are even functions due to the derivation properties, B and C are odd functions. A matrix of this block structure is called even matrix.

The transformation factor itself depends on the oriented Berezinian of the Jacobian. This is defined as:Ber_{+-} J_{alphaeta} = sgn, operatorname{det} A, operatorname{Ber} J_{alphaeta}

For further details see the article about the Berezinian.

The complete formula now reads as: int_U f(x_a, heta_i) d(x, heta) =int_{U'} f(x_a, heta_i) operatorname{Ber}_{+-}, frac{partial(x_a, heta_i)}{partial(y_b,xi_j)} d(y,xi)

Literature

* Theodore Voronov: "Geometric integration theory on Supermanifolds", Harwood Academic Publisher, ISBN 3-7186-5199-8

References


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