Abstract
We will proceed to describe how changes of variables effects the integration measure for odd variables. We will do this via a simple example rather than in full generality.

Integration measure with two odd variables
Let us consider the integration with respect to two odd variables, \(\{ \theta, \overline{\theta} \}\). Let us consider a change in variables of the form

\(\theta^{\prime} = a \theta + b \overline{\theta}\),
\( \overline{\theta}^{\prime} = c \theta + d \overline{\theta}\),

where a,b,c,d are real numbers (or complex if you wish).

Now, one of the basic properties of integration is that it should not depend on how you parametrise things. In other worlds we get the same result whatever variables we chose to employ. For the example at hand we have

\( \int D(\overline{\theta}^{\prime}, \theta^{\prime}) \theta^{\prime} \overline{\theta}^{\prime} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

Thus, we have

\(\int D(\overline{\theta}^{\prime}, \theta^{\prime}) (ad-bc)\theta \overline{\theta} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

In order to be invariant we must have

\(\int D(\overline{\theta}^{\prime}, \theta^{\prime})= \frac{1}{(ad-bc) }D(\overline{\theta}, \theta) \).

Note that the factor (ad-bc) is the determinant of a 2×2 matrix. However, note that we divide by this factor and not multiply in the above law. This is a general feature of integration with respect to odd variables, one divides by the determinant of the transformation matrix rather than multiply. This generalises to non-linear transformations that mix even and odd coordinates on a supermanifold. This is the famous Berezinian. A detailed discussion is outside the remit of this introduction.

Furthermore, note that the transformation law for the measure is really the same as the transformation law for derivatives. Thus, the Berezin measure is really a mixture of algebraic and differential ideas.

What next?
I think this should end our discussion of the elementary properties of analysis with odd variables. I hope it has been useful to someone!

Abstract
We now proceed to define integration with respect to odd variables.

The fundamental theorem of calculus for odd variables
Let us consider just one odd variable. This will be sufficient for our purposes for now. Following the direct analogy with integration of functions over a circle the fundamental theorem of calculus states

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

We use the notation \(D\theta \) for the measure rather than \(d\theta \) as the measure cannot be associated with a one-form. We will discuss this in more detail another time.

Definition of integration
Recall that the general form of a function in one odd variable is

\(f(\theta) = a + \theta b\),

with a and b being real numbers. Thus from the fundamental theorem we have

\(\int D\theta b =0\).

In particular this implies

\(\int D\theta =0\).

Then we have

\(\int D\theta f(\theta) = a \int D\theta + b \int D\theta \:\: \theta = b \int D\theta\:\: \theta \).

Thus to define integration all we have to do is define the normalisation

\(\int D\theta\:\: \theta\).

The choice made by Berezin was to set this to unity. Other choices are also just as valid. Thus,

\(\int D\theta f(\theta) = b\).

Integration for several odd variables
For the case of more than one odd variable one simply uses

\(\int D(\theta_{1}, \theta_{2} , \cdots \theta_{n})f(\theta) = \int D\theta_{1} \int D \theta_{2} \cdots \int D\theta_{n} f(\theta)\).

example Consider two odd variables.

\(\int D(\overline{\theta}, \theta) \left( f_{0} + \theta \:f + \overline{\theta}\: \overline{f} + \theta \overline{\theta}F \right) = F \).

The general rule is that (taking care with signs) the integration with respect to the measure \(D(\theta_{1}, \theta_{2} , \cdots \theta_{n})\) of a function picks out the coefficient of the \(\theta_{1}, \theta_{2} , \cdots \theta_{n}\) term.

Integration and differentiation are the same!
From the above we see that differentiation with respect to an odd variable is the same as integration with respect to the odd variable. This explains why we cannot associate a “top-form” with the measure. This will become more apparent when we discuss changes of variables.

What next?
Next we will examine how changing variables in the integration effects the measure. We will see that things look “upside down” as compared with the integration of real variables. This is anticipated by the equivalence of integration and differentiation.

Abstract
Before we consider odd variables, let us describe how to algebraically define integration of functions over the circle.

Functions on the circle
Recall the Fourier expansion. It is well known that any continuous function on the circle is of the form

\(f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right) \),

with the a’s and b’s being constants, i.e. independent of the variable x.

The fundamental theorem of calculus
The fundamental theorem of calculus states that

\(\int_{S^{1}} dx \: \frac{\partial f(x)}{\partial x } = 0 \),

as functions on the circle are periodic.

Integration of functions
It turns out that integration of functions over the circle can be defined algebraically up to a choice in measure. To see this observe

\(\int_{S^{1}} dx f(x) = \int_{S^{1}} dx \frac{a_{0}}{2} + \int_{S^{1}} dx \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right)\)

Then we can write

\(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx + \int_{S_{1}} dx \frac{\partial }{\partial x} \sum_{n=1}^{\infty} \left ( \frac{a_{n}}{n}\sin(nx) + \frac{- b_{n}}{n} \cos(nx) \right)\)

to get via the fundamental theorem of calculus

\(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx\).

So we have just about defined integration completely algebraically from the fundamental theorem of calculus. All we have to do is specify the normalisation

\(\int_{S^{1}} dx \).

The standard choice would be

\(\int_{S^{1}} dx = 2 \pi\),

to get back to our usual notion of integration of periodic functions. Though it would be quite consistent to consider some other normalisation, say to unity.

Anyway, up to a normalisation the integration of functions over the circle selects the “constant term” of the corresponding Fourier expansion.

What next?
So, the above construction demonstrates that integration of functions over a domain without boundaries can be defined algebraically, up to a normalisation. This served as the basis for Berezin who defined the notion of integration of odd variables.

Recall that odd variables have no topology and no boundaries. The integration with respect to such variables cannot be in the sense of Riemann. However, thinking of functions of odd variables in analogy to periodic functions integration can be defined algebraically. We will describe this next time.

Abstract
Here we will define the notion of differentiation with respect to an odd variable and examine some basic properties.

Definition
Differentiation with respect to an odd variable is completely and uniquely defined via the following rules:

1. \(\frac{ \partial \theta^{\beta} }{\partial \theta^{\alpha}} = \delta_{\alpha}^{\beta} \).
2. Linearity:
   \(\frac{\partial}{\partial \theta }(a f(\theta)) = a \frac{\partial}{\partial \theta } f(\theta)\).
   \(\frac{\partial}{\partial \theta }( f(\theta) + g(\theta)) = \frac{\partial}{\partial \theta }f(\theta) + \frac{\partial}{\partial \theta } g(\theta)\).
3. Leibniz rule:
   \(\frac{\partial}{\partial \theta }( f(\theta)g(\theta)) = \frac{\partial f(\theta)}{\partial \theta } + (-1)^{\widetilde{f}} f \frac{\partial g(\theta)}{\partial \theta } \).

The operator \(\frac{\partial }{\partial \theta }\) is odd, that is it changes the parity of the function it acts on. This must be taken care of when applying Leibniz’s rule.

Elementary properties
It is easy to see that

\(\frac{\partial}{\partial \theta^{\alpha}}\frac{\partial}{\partial \theta^{\beta}}+ \frac{\partial}{\partial \theta^{\beta}}\frac{\partial}{\partial \theta^{\alpha}}=0\),

in particular

\(\left( \frac{\partial}{\partial \theta} \right)^{2}=0\).

Example
\(\frac{\partial}{\partial \theta} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = b + \overline{\theta}d\).

Example
\(\frac{\partial}{\partial \overline{\theta}} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = c- \theta d\).

Changes of variables
Under changes of variable of the form \(\theta \rightarrow \theta^{\prime}\) the derivative transforms as standard

\(\frac{\partial}{\partial \theta^{\prime}} = \frac{\partial\theta}{\partial \theta^{\prime}} \frac{\partial}{ \partial \theta}\).

We will have a lot more to say about changes of variables (coordinates) another time.

What next?
We now know how to define and use the derivative with respect to an odd variable. Note that this was done algebraically with no mention of limits. As the functions in odd variables are polynomial the derivative was simple to define.

Next we will take a look at integration with respect to an odd variable. We cannot think in terms of boundaries, limits or anything resembling the Riemann or Lebesgue notions of integration. Everything will need to be done algebraically.

This will lead us to the Berezin integral which has the strange property that integration and differentiation with respect to an odd variable are the same.

Abstract
Here we will present the very basic ideas of Grassmann variables and polynomials over them.

Grassmann algebra
Consider a set of n odd variables \(\{ \theta^{1}, \theta^{2}, \cdots \theta^{n} \}\). By odd we will mean that they satisfy

\( \theta^{a}\theta^{b} + \theta^{b} \theta^{a}=0\).

Note that in particular this means \(\theta^{2}=0\). That is the generators are nilpotent.

The Grassmann algebra is then defined as the polynomial algebra in these variables. Thus a general function in odd variables is

\(f(\theta) = f_{0} + \theta^{a}f_{a} + \frac{1}{2!} \theta^{a} \theta^{b}f_{ba} + \cdots + \frac{1}{n!} \theta^{a_{1}} \cdots \theta^{a_{n}}f_{a_{n}\cdots a_{1}}\).

The coefficients we take as real and antisymmetric. Note that the nilpotent property of the odd variables means that the Grassmann algebra is complete as polynomials.

Example If we have the algebra generated by a single odd variable \(\theta \) then polynomials are of the form

\(a + \theta b\).

Example If we have two odd variables \(\theta\) and \(\overline{\theta}\) then polynomials are of the form

\(a + \theta b + \overline{\theta} c + \theta \overline{\theta} d\).

It is quite clear that the polynomials in odd variables forms a vector space. You can add such functions and multiply by a real number and the result remains a polynomial. It is also straightforward to see that we have an algebra. One can multiply two such functions together and get another.

The space of all such functions has a natural \(\mathbb{Z}_{2}\)-grading, which we will call parity given by the number of odd generators in each function mod 2. If the function has an even/odd number of odd variables then the function is even/odd. We will denote the parity by of a function \(\widetilde{f}= 0/1\), if it is even/odd.

Example \(a +\theta \overline{\theta} d \) is an even function and \(\theta b + \overline{\theta} c \) is an odd function.

Let us define the (super)commutator of such functions as

\([f,g] = fg -(-1)^{\widetilde{f} \widetilde{g}} gf\).

If the functions are not homogeneous, that is even or odd the commutator is extended via linearity. We see that the commutator of any two functions in odd variables vanishes. Thus we say that the algebra of functions in odd variables forms a supercommutative algebra.

Specifically note that this means the ordering of odd functions is important.

Superspaces
The modern approach to geometry is to define and deal with “spaces” in terms of the functions upon them. Geometrically we can think of the algebra generated by n odd variables as defining the space \(\mathbb{R}^{0|n}\). Note that no such “space” in the classical sense exists. In fact such spaces consist of only one point!

If we promote the coefficients in the polynomials to be functions of m real variables then we have the space \(\mathbb{R}^{m|n}\). We are now most of the way to defining supermanifolds, but this would be a digression from the current issues.

Noncommutative superalgebras
Of course superalgebras for which the commutator generally is non-vanishing can be defined and are naturally occurring. We will encounter such things when dealing with first order differential operators acting on functions in odd variables. Geometrically these are the vector fields. Recall that the Lie bracket between vector fields over a manifold is in general non-vanishing.

What next?
Given the basic algebraic properties of functions in odd variables we will proceed to algebraically define how to differentiate with respect to odd variables.

Forward
Following a conversation on a popular science chat room the subject of Grassmann variables and in particular the Berezin integral arose. Thus I decided to with a short introduction to the basic theory of superalgebras, particularly supercommutative algebras and their calculus.

We will be primarily interested in algebras that involve the Koszul sign rule, that is include an extra minus sign when you interchange odd elements:

\(ab = – ba\).

Ancient History
The beginning of all supermathematics can be traced back to 1885 and the work of Hermann Günther Grassmann on linear algebra. He introduced variables that involve a minus sign when interchanging their order. Élie Cartan’s theory of differential forms is also in hindsight a “super-theory”. Many other constructions in algebra and topology can be thought of as “super” and involve a sign factor when interchanging the order.

Physics
By the early 1950’s odd variables appeared in quantum field theory as a semiclassical description of fermions. Initially the analysis was based on the canonical description of quantisation and so confined to derivatives with respect to odd variables. Berezin in 1961 introduced the integration theory for odd variables and this was promptly applied to the path integral approach to quantisation.

Supermanifolds
In these early works odd variables were understood very formally in an algebraic way. That is they were not associated with with any general notion of a space. Berezin’s treatment of even and odd variables convinced him that there should be a way to treat them analogously to real and complex variables in complex geometry. The bulk of this work was carried out by Berezin and his collaborators between 1965 and 1975. Berezin introduced general non-linear transformations that mix even and odd variables as well as generalisation of the determinant to integration over even and odd variables. This work led to the notion of superspaces and supermanifolds. In essence one thinks of a supermanifold as a “manifold” with even (commuting) and odd (anticommuting) coordinates. A detailed discussion of supermanifolds is out of the scope of this introduction.

Supersymmetric field theories
The nomenclature super comes from physics. Gol’fand & Likhtman extended the Poincare group to include “odd translations”. These operators are fermionic in nature and thus require anticommutators in the extended Poincare algebra. Supersymmetry is a remarkable symmetry that mixed bosonic and fermionic degrees of freedom. Lagrangians (or actions) that exhibit supersymmetry have some very attractive features. The surprising result is that supersymmetry can cancel most or even all of the divergences of certain quantum field theories. A detailed discussion of supersymmetric field theories is outside the scope of this introduction.

Gauge theories and the BRST symmetry
The use of odd variables is also necessary in (perturbative) non-abelian gauge theories (in the covariant gauges at least), even if one initially restricts attention to theories without fermions. There are several complications that do not arise in abelian gauge theory. These originate primarily from the gauge fixing, which effects the path integration measure in a non-trivial way. Feynman in 1963 showed that using standard quantisation methods available at the time, Yang-Mills theory was not unitary. Feynman also showed that counter terms, now known as ghosts could be added that remove the nonunitary parts. Originally these ghost, which are odd but violate the spin-statistics theorem were seen as ad-hoc. Later Faddeev and Popov showed that these ghost arise in the theory by considering the so called Faddeev-Popov determinant.

It was noticed that the gauge fixed Lagrangian possess a new global (super)symmetry that rotates the gauge fields into ghosts. This symmetry is named after it’s discoverers Becchi, Rouet, Stora and independently Tyutin, thus BRST symmetry. As this is a global symmetry no new degrees of freedom can be eliminated.

The BRST symmetry is now a fundamental tool when dealing with quantum gauge theories. For example the BRST symmtery is important when considering the remormalisability and absence of anomalies for a given theory. We will not say any more about gauge theories in this introduction.

Mathematical applications
Odd elements can be employed very successfully in pure mathematics. For example, the de Rham complex of a manifold can be completely understood in terms of functions and vector fields over a particular supermanifold. Multivector fields can also be thought of in a similar way in terms of a supermanifold and an odd analogue of a Poisson bracket.

Various algebraic structures can be encoded in superalgebras that come equipped with a homological vector field. That is an odd vector field that “squares to zero”

\(Q^{2} = \frac{1}{2}[Q,Q]=0\).

Common examples include Lie algebras, \(L_{\infty}\)-algebras, Lie algebroids, \(A_{\infty}\)-algebra etc.

Guide to this introduction
I hope that these opening words have convinced you that the study of superalgebras and Grassmann odd variables is useful in physics and pure mathematics.

I will be quite informal in presentation and attitude. The intention is to convey the main ideas without over burdening the reader.

A tentative guide is as follows:

1. Elementary algebraic properties of superalgebras.
2. Differential calculus of odd variables.
3. Integration with respect to odd variables: the Berezin integral.

Quick guide to references
The mathematical theory of Grassmann algebras, superalgebras and supermanifolds is well established and can be found in several books. Any book on quantum field theory will say something about the algebra and calculus of odd variables. The mathematical books that I like include:

• Gauge Field Theory and Complex Geometry, Yuri I. Manin, Springer; 2nd edition (June 27, 1997).
• Geometric Integration Theory on Supermanifolds, Th. Th. Voronov, Routledge; 1 edition (January 1, 1991).
• Supersymmetry for Mathematicians: An Introduction, V. S. Varadarajan, American Mathematical Society (July 2004).

Other books that deserve a mention are

• Supermanifolds, Bryce DeWitt, Cambridge University Press; 2 edition (June 26, 1992).
• Supermanifolds: Theory and Applications, A. Rogers, World Scientific Publishing Company (April 18, 2007).