Geometric meaning of Berezin integration
Berezin integration in a Grassmann algebra is defined such that its algebraic properties are analogous to definite integration of ordinary functions: linearity (taking anticommutativity into account), scale invariance and independence from the integration variable. Up to normalization this means that
$$\int f(\theta) d\theta = \int (f_0 + f_1\theta) d\theta = f_1$$
I already have a good geometric picture of Grassmann numbers themselves as elements of an exterior algebra, but I've never understood the geometric meaning of Berezin integration. What exactly are we "summing" and over what domain? Why the strange scaling property $d(a\theta) = a^{-1} d\theta$?
Solution 1:
I am quite sure that Berezin integral cannot be interpreted as a sum in any sense. But one can find a geometric picture of Berezin integral.
How should one think about supermanifold geometrically? One way to think about manifold is following. It is a topological space and sheaf of functions (i.e. for each open subset one assign a ring). You may want to consider smooth, complex, algebraic etc manifold. Then those functions should be $C^{\infty}$, holomorphic or algebraic respectively. (By the way, there are some conditions... you cannot take topological space to be a point and assign a ring of polynomials.)
For supermanifold rings (assigned to each open subspace) are supercommutative (i.e. $\mathbb{Z}/2\mathbb{Z}$ graded with $ab=(-1)^{|a| |b|} ba$). So the point is that if one factor over elements of degree 1, then it is an ordinary manifold. This manifold is called underlying manifold.
This picture recalls a standard picture from algebraic geometry about non-reduced scheme. Let me briefly remind you using an example. Consider intersection of circle $x^2 + y^2 = 1$ and a line $y=1$. This intersection is a point. On the other hand $\mathbb{k}[x,y]/(y-1, x^2+y^2-1) = \mathbb{k}[x]/(x^2)$. So the ring of functions is not $\mathbb{k}$ (as it should be for point). This is a double point. Also one can think about this double point as a fusion of two regular points. To sum up, in ordinary algebraic geometry there may emerge nilpotents in a ring of functions. One should think about them as infinitesimal thickening of our initial variety.
The same happens with supermanifold. Note that all these odd coordinates are nilpotent. However, I would not recommend you to think about this odd direction as thickening. For example, correct notion of dimension of supermanifold is a number of even coordinates minus number of odd coordinates. I will try to justify this mysterious concept later.
When I try to explain this concept to a physicist, I say that odd coordinates do not correspond to any physical direction. That is why Berezin integral is not a sum.
Construction of supermanifold Let $M$ be a manifold an $E$ be a vector bundle over $M$. Let us define a supermanifold $\Pi E$. As topological space, it is homeomorphic to $M$. But the ring of functions is sections of $\Lambda^* (E^*)$. So locally there are $\dim M$ even coordinates and $\text{rank} E$ odd coordinates.
Theorem Any supermanifold is (not canonically) isomorphic to $\Pi E$ for some $E$.
Canonical measure on $\Pi T M$. The functions on this space is are differential forms $\theta_i = dx_i$. One can integrate a differential form of highest rank over $M$. We will see that this correspond to canonical measure.
One should think about $d \theta_i$ as a vector field since $\int \theta_j d \theta_i = \delta_{ij}$ (so $d \theta_i$ is dual basis to $\theta_i$). So $d \theta_1 \dots \theta_n dx_1 \dots dx_n$ is canonically defined.
Let $F(x, \theta)= f_0(x) + \dots + f_n(x) \theta_1 \dots \theta_n$ Then $$\int_{\Pi TM} F(x, \theta) d \theta_1 \dots d\theta_n dx_1 \dots dx_n = \int_{M} f_n(x) dx_1 \dots dx_n$$ So this is standart integration of form of highest rank! This picture justifies dimension formula since canonical measure can appear only on a zero-dimensional manifold (do not be too strict to this argument).
Berezin measure. Since any supermanifold is isomorphic to $\Pi E$, let us think about Berezin measure on $\Pi E$. It is section of $\Lambda E^* \otimes \Omega^{\dim M} M$. Integration is following procedure. First of all one use pairing $\Lambda^r E^* \otimes \Lambda^r E \rightarrow \mathbb{k}$. So Berezin measure defines a map
$$\Gamma ( \Lambda^r E ) \rightarrow \Omega^n M $$ Then one just integrate differential form. This procedure explains geometric meaning of Berezin integral in terms of classical (not super) geometry.
Solution 2:
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User AccidentalFourierTransform has already given a good answer. He is exactly right that Berezin integration $$\int\! d\theta= \frac{\partial }{\partial \theta}\tag{A} $$ is the same as differentiation, which explains the strange scaling property that OP noticed.
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I explain in the first part of my Phys.SE answer here why definition (A) is unique (up to an overall multiplicative normalization factor).
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One thing that one may wonder about is whether it makes sense to define a definite integral$^1$ $$I(\theta_1,\theta_2):=\int_{\theta_1}^{\theta_2}\! d\theta\tag{B}$$ over a Grassmann-interval $[\theta_1,\theta_2]$? I investigate this in the second part of my Phys.SE answer here. While I stop short of claiming a no-go theorem, the various possibilities are rather un-appealing/not useful.
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In particular, it is not useful to think of Berezin integration (A) over the odd line $\mathbb{R}^{0|1}$ as composed of some sort of sum over Grassmann-intervals, cf. OP's question. It's a purely algebraic construction, cf. eq. (A).
References:
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Pierre Deligne and John W. Morgan, Notes on Supersymmetry (following Joseph Bernstein). In Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, American Mathematical Society (1999) 41–97.
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V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, 2004.
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$^1$ How the Grassmann-odd indeterminates $\theta$, $\theta_1$, $\theta_2$ are actually defined within the theory of supermanifolds is a question in its own right, cf. e.g. Refs. 1 & 2. In this answer, we will just need some of their basic properties.