Papyrus 61

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In mathematical physics, a Berezin integral, named after Felix Berezin, (or Grassmann integral, after Hermann Grassmann) is a way to define integration of elements of the exterior algebra (Hermann Grassmann 1844). It is called integral because it is used in physics as a sum over histories for fermions, an extension of the path integral.

Integration on an exterior algebra

Let Λn be the exterior algebra of polynomials in anticommuting elements θ1,,θn over the field of complex numbers. (The ordering of the generators θ1,,θn is fixed and defines the orientation of the exterior algebra.) The Berezin integral on Λn is the linear functional Λndθ with the following properties:

Λnθnθ1dθ=1,
Λnfθidθ=0,i=1,,n

for any fΛn, where /θi means the left or the right partial derivative. These properties define the integral uniquely. The formula

Λnf(θ)dθ=Λ1(Λ1(Λ1f(θ)dθ1)dθ2)dθn

expresses the Fubini law. On the right-hand side, the interior integral of a monomial f=g(θ)θ1 is set to be g(θ), where θ=(θ2,...,θn); the integral of f=g(θ) vanishes. The integral with respect to θ2 is calculated in the similar way and so on.

Change of Grassmann variables

Let θi=θi(ξ1,...,ξn),i=1,...,n, be odd polynomials in some antisymmetric variables ξ1,...,ξn. The Jacobian is the matrix

D={θiξj,i,j=1,...,n},

where the left and the right derivatives coincide and are even polynomials. The formula for the coordinate change reads

f(θ)dθ=f(θ(ξ))(detD)1dξ.

Berezin integral

Consider now the algebra Λmn of functions of real commuting variables x=x1,...,xm and of anticommuting variables θ1,...,θn (which is called the free superalgebra of dimension (mn)). This means that an element f=f(x,θ)Λmn is a function of the argument x that varies in an open set Xm with values in the algebra Λn. Suppose that this function is continuousand vanishes in the complement of a compact set Km. The Berezin integral is the number

Λmnf(x,θ)dθdx=mdxΛnf(x,θ)dθ.

Change of even and odd variables

Let a coordinate transformation be given by xi=xi(y,ξ),i=1,...,m;θj=θj(y,ξ),j=1,...,n, where xi,yi are even and θj,ξj are odd polynomials of ξ depending on even variables y. The Jacobian matrix of this transformation has the block form:

J=(x,θ)(y,ξ)=(ABCD),

where each even derivative /yj commutes with all elements of the algebra Λmn; the odd derivatives commute with even elements and anticommute with odd elements. The entries of the diagonal blocks A=x/y and D=θ/ξ are even and the entries of the offdiagonal blocks B=x/ξ,C=θ/y are odd functions, where /ξj mean right derivatives. The Berezinian (or the superdeterminant) of the matrix J is the even function

BerJ=det(ABD1C)detD1

defined when the function detD is invertible in Λmn. Suppose that the real functions xi=xi(y,0) define a smooth invertible map F:YX of open sets X,Y in m and the linear part of the map ξθ=θ(y,ξ) is invertible for each yY. The general transformation law for the Berezin integral reads

Λmnf(x,θ)dθdx=Λmnf(x(y,ξ),θ(y,ξ))εBerJdξdy
=Λmnf(x(y,ξ),θ(y,ξ))εdet(ABD1C)detDdξdy,

where ε=sgndetx(y,0)/y is the sign of the orientation of the map F. The superposition f(x(y,ξ),θ(y,ξ)) is defined in the obvious way, if the functions xi(y,ξ) do not depend on ξ. In the general case, we write xi(y,ξ)=xi(y,0)+δi, where δi,i=1,...,m are even nilpotent elements of Λmn and set

f(x(y,ξ),θ)=f(x(y,0),θ)+ifxi(x(y,0),θ)δi+12i,j2fxixj(x(y,0),θ)δiδj+...,

where the Taylor series is finite.

History

The mathematical theory of the integral with commuting and anticommuting variables was invented and developed by Felix Berezin. Some important earlier insights were made by David John Candlin. Other authors contributed to these developments, including the physicists Khalatnikov [3] (although his paper contains mistakes), Matthews and Salam [4], and Martin [6].

See also

References

[1] F.A. Berezin, The Method of Second Quantization, Academic Press, (1966)

[2] F.A. Berezin, Introduction to superanalysis. D. Reidel Publishing Co., Dordrecht, 1987. xii+424 pp. ISBN 90-277-1668-4.

[3] I.M. Khalatnikov (1954), "Predstavlenie funkzij Grina v kvantovoj elektrodinamike v forme kontinualjnyh integralov" (Russian). JETP, 28, 635.

[4] P.T. Matthews, A. Salam (1955), "Propagators of quantized field". Nuovo Cimento 2, 120.

[5] D.J. Candlin (1956)."On Sums over Trajectories for Systems With Fermi Statistics". Nuovo Cimento 4:231. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park..

[6] J.L. Martin (1959), "The Feynman principle for a Fermi System". Proc. Roy. Soc. A 251, 543.