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Ordered Berezin integral detects a sign in a permuted top monomial

Example

In the ordered algebra with generators $θ^{1}, θ^{2}, {\overset{ˉ}{θ}}^{\dot{1}}, {\overset{ˉ}{θ}}^{\dot{2}}$ and measure $d^{4} θ = d {\overset{ˉ}{θ}}^{\dot{2}} d {\overset{ˉ}{θ}}^{\dot{1}} d θ^{2} d θ^{1}$ , let

$f = c θ^{1} {\overset{ˉ}{θ}}^{\dot{1}} θ^{2} {\overset{ˉ}{θ}}^{\dot{2}} + d θ^{1} θ^{2} {\overset{ˉ}{θ}}^{\dot{1}} {\overset{ˉ}{θ}}^{\dot{2}}, c, d \in C .$

Then $\int d^{4} θ f = d - c$ .

Example references

Martin, §4.1.

Gates et al., §3.7.

Four-dimensional $N = 1$ superspace Grassmann algebra and Berezin integration: Four-dimensional $N = 1$ superspace Grassmann algebra and Berezin integration.

Associativity and coefficient calculus for the finite superspace Grassmann algebra: Associativity and coefficient calculus for the finite superspace Grassmann algebra.

Facts & Assumptions

Given: The ordered four-generator algebra, the displayed polynomial, and the ordered four-variable coefficient-extraction measure.

[F1]

Four-dimensional $N = 1$ superspace Grassmann algebra and Berezin integration: $θ^{i} θ^{j} = - θ^{j} θ^{i}$ for distinct ordered generators, and the measure extracts the coefficient of $θ^{1} θ^{2} {\overset{ˉ}{θ}}^{\dot{1}} {\overset{ˉ}{θ}}^{\dot{2}}$ .

[F2]

Associativity and coefficient calculus for the finite superspace Grassmann algebra: the signed basis product is associative and the ordered four-variable integral is a well-defined linear functional equal to top-coefficient extraction.

Verification

Given: [F1], [F2], and the polynomial in the Example.

1.1F1algebra

In the first monomial, moving $θ^{2}$ left past ${\overset{ˉ}{θ}}^{\dot{1}}$ makes one transposition and contributes a minus sign. Thus $θ^{1} {\overset{ˉ}{θ}}^{\dot{1}} θ^{2} {\overset{ˉ}{θ}}^{\dot{2}} = - θ^{1} θ^{2} {\overset{ˉ}{θ}}^{\dot{1}} {\overset{ˉ}{θ}}^{\dot{2}}$ , so the ordered top coefficient of $f$ is $- c + d$ .

2.1F1F2step 1.1∎

By [F2], the ordered measure returns exactly that top coefficient; hence $\int d^{4} θ f = d - c$ . The result depends on the stated measure order and generator convention.

Ordered Berezin integral detects a sign in a permuted top monomial

Example

Example references

Facts & Assumptions

Verification

Depends on

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Sources