Grassmann integral

In mathematical physics, a Grassmann integral, or, more correctly, 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 because it has analogous properties and since it is used in physics as a sum over histories for fermions, an extension of the path integral. The technique was invented by the Russian mathematician Felix Berezin and developed in his textbook.^[1] Some earlier insights were made by the physicist David John Candlin^[2] in 1956.

Definition

The Berezin integral is defined to be a linear functional

\int [af(\theta)+bg(\theta)] \, d\theta = a\int f(\theta) \, d\theta + b\int g(\theta) \, d\theta

where we define

\int \theta \, d\theta = 1

\int \, d\theta = 0

so that :

\int \frac\partial{\partial\theta}f(\theta)\,d\theta = 0.

These properties define the integral uniquely.

\int (a\theta+b)\, d\theta = 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(\theta_1)\cdots f_n(\theta_n)\, d\theta_1\cdots \, d\theta_n = \int f_1(\theta_1) \, d\theta_1\cdots \int f_n(\theta_n) \, d\theta_n.

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

Suppose now we want to do a substitution:

\theta_i=\theta_i(\xi_j)

where as usual (ξ_j) implies dependence on all ξ_j. Moreover the function θ_i has to be an odd function, i.e. contains an odd number of ξ_j in each summand. The Jacobian is the usual matrix

J_{ij}=\frac{\partial\theta_i}{\partial\xi_j}.

the substitution formula now reads as

\int f(\theta_i) \, d\theta = \int f(\theta_i(\xi_j)) \det(J_{ij})^{-1} \, d\xi.

Substitution formula

Consider now a mixture of even and odd variables, i.e. x_a and θ_i. Again we assume a coordinate transformation as $x_a=x_a(y_b,\xi_j)\,,\; \theta_i=\theta_i(y_b,\xi_j)\;,$ where x_a are even functions and θ_i are odd functions. We assume the functions x_a and θ_i to be defined on an open set U in R^m. The functions x_a map onto the open set U' in R^m.

The change of the integral will depend on the Jacobian

J_{\alpha\beta}=\frac{\partial (x_a,\theta_i)}{\partial(y_b,\xi_j)}.

This matrix consists of four blocks:

J=\begin{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:

\operatorname{Ber}_{+-} J_{\alpha\beta} = \operatorname{sgn}\, \operatorname{det} A\, \operatorname{Ber} J_{\alpha\beta}.

For further details see the article about the Berezinian.

The complete formula now reads as:

\int_U f(x_a,\theta_i) \, d(x,\theta) = \int_{U'} f(x_a,\theta_i) \operatorname{Ber}_{+-}\, \frac{\partial(x_a,\theta_i)}{\partial(y_b,\xi_j)} \, d(y,\xi).

Gaussian integrals over Grassmann variables

The following formulas for Gaussian integrals are used often in the path integral formulation of quantum field theory:

\int \exp\left[\theta^TA\eta\right] \,d\theta\,d\eta = \det A

with $A$ being a $n \times n$ matrix.

\int \exp\left[-\theta^T M \theta\right] \,d\theta = \begin{cases} 2 ^{n\over 2 } \sqrt{ \det M } , & n \mbox{ even} \\ 0 , & n \mbox{ odd} \end{cases}

with $M$ being an $n \times n$ antisymmetric matrix. In the above formulas the notation $d \theta = d\theta_1\cdots \, d\theta_n$ is used.

From the above formulas, other useful formulas follow:

\int \exp\left[\theta^TA\eta +\theta^T J + K^T \eta \right] \,d\theta\,d\eta = \det A \,\,\exp[-K^T A^{-1} J ]

with $A$ being an invertible $n \times n$ matrix, $K$ and $J$ are Grassmann variables.

\int \exp\left[-{1\over 2} \theta^T M \theta + \eta^T \theta \right] \,d\theta = \begin{cases} \sqrt{ \det M } \exp \left[ -{1\over 2} \eta^T M^{-1} \eta \right] , & n \mbox{ even} \\ 0 , & n \mbox{ odd} \end{cases}

with $M$ being an invertible $n \times n$ antisymmetric matrix, and $\eta$ Grassmann variable.

Literature

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

References

↑ A. Berezin, The Method of Second Quantization, Academic Press, (1966)
↑ D.J. Candlin (1956). "On Sums over Trajectories for Systems With Fermi Statistics". Nuovo Cimento 4: 231. doi:10.1007/BF02745446.