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

[math]\displaystyle{ \int [af(\theta)+bg(\theta)] \, d\theta = a\int f(\theta) \, d\theta + b\int g(\theta) \, d\theta }[/math]

where we define

[math]\displaystyle{ \int \theta \, d\theta = 1 }[/math]

[math]\displaystyle{ \int \, d\theta = 0 }[/math]

so that :

[math]\displaystyle{ \int \frac\partial{\partial\theta}f(\theta)\,d\theta = 0. }[/math]

These properties define the integral uniquely.

[math]\displaystyle{ \int (a\theta+b)\, d\theta = a. }[/math]

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:

[math]\displaystyle{ \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. }[/math]

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

Suppose now we want to do a substitution:

[math]\displaystyle{ \theta_i=\theta_i(\xi_j) }[/math]

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

[math]\displaystyle{ J_{ij}=\frac{\partial\theta_i}{\partial\xi_j}. }[/math]

the substitution formula now reads as

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

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 [math]\displaystyle{ x_a=x_a(y_b,\xi_j)\,,\; \theta_i=\theta_i(y_b,\xi_j)\;, }[/math] 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

[math]\displaystyle{ J_{\alpha\beta}=\frac{\partial (x_a,\theta_i)}{\partial(y_b,\xi_j)}. }[/math]

This matrix consists of four blocks:

[math]\displaystyle{ J=\begin{bmatrix}A&B\\ C&D \end{bmatrix}. }[/math]

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:

[math]\displaystyle{ \operatorname{Ber}_{+-} J_{\alpha\beta} = \operatorname{sgn}\, \operatorname{det} A\, \operatorname{Ber} J_{\alpha\beta}. }[/math]

For further details see the article about the Berezinian.

The complete formula now reads as:

[math]\displaystyle{ \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). }[/math]

Gaussian integrals over Grassmann variables

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

[math]\displaystyle{ \int \exp\left[-\theta^TA\eta\right] \,d\theta\,d\eta = \det A }[/math]

with [math]\displaystyle{ A }[/math] being a [math]\displaystyle{ n \times n }[/math] matrix.

[math]\displaystyle{ \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} }[/math]

with [math]\displaystyle{ M }[/math] being an [math]\displaystyle{ n \times n }[/math] antisymmetric matrix. In the above formulas the notation [math]\displaystyle{ d \theta = d\theta_1\cdots \, d\theta_n }[/math] is used.

From the above formulas, other useful formulas follow:

[math]\displaystyle{ \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 ] }[/math]

with [math]\displaystyle{ A }[/math] being an invertible. Note that these integrals are all in the form of a partition function.

See also

• Berezin integral
• Supermanifold

Literature

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

References

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