In mathematical physics, a Grassmann number, named after Hermann Grassmann, (also called an anticommuting number) is an element of the exterior algebra over complex numbers. Grassman numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed.
Grassman numbers are generated by anti-commuting elements or objects. The idea of anti-commuting objects arises in multiple areas of mathematics: they are typically seen in differential geometry, where the differential forms are anti-commuting. Differential forms are normally defined in terms of derivatives on a manifold; however, one can contemplate the situation where one "forgets" or "ignores" the existence of any underlying manifold, and "forgets" or "ignores" that the forms were defined as derivatives, and instead, simply contemplate a situation where one has objects that anti-commute, and have no other pre-defined or pre-supposed properties. Such objects form an algebra, and specifically the Grassman algebra or exterior algebra.
The Grassman numbers are elements of that algebra. The appelation of "number" is justified by the fact that they behave not unlike "ordinary" numbers: they can be added, multiplied and divided: they form a field. More can be done: one can consider polynomials of Grassman numbers, leading to the idea of holomorphic functions. One can take derivatives of such functions, and then consider the anti-derivatves as well. Each of these ideas can be carefully defined, and correspond reasonably well to the equivalent concepts from ordinary mathematics. The analogy does not stop there: one has an entire branch of supermathematics, where the analog of Euclidean space is superspace, the analog of a manifold is a supermanifold, the analog of a Lie algebra is a Lie superalgebra and so on. The Grassman numbers are the underlying construct that make this all possible.
Of course, one could pursue a similar program for any other field, or even ring, and this is indeed widely and commonly done in mathematics. However, supermathematics takes on a special significance in physics, because the anti-commuting behavior can be strongly identified with the quantum-mechanical behavior of fermions: the anti-commutation is that of the Pauli exclusion principle. Thus, the study of Grassman numbers, and of supermathematics, in general, is strongly driven by their utility in physics.
Specifically, in quantum field theory, or more narrowly, second quantization, one works with ladder operators that create multi-particle quantum states. The ladder operators for fermions create field quanta that must necessarily have anti-symmetric wave functions, as this is forced by the Pauli exclusion principle. In this situation, a Grassman number corresponds immediately and directly to a wave function that contains some (typically indeterminate) number of fermions.
Grassman numbers are individual elements or points of the exterior algebra generated by a set of n Grassman variables with n possibly being infinite. The usage of the term "Grassman variables" is historic; they are not variables, per se; they are better understood as the basis elements of a unital algebra. The terminology comes from the fact that a primary use is to define integrals, and that the variable of integration is Grassman-valued, and thus, by abuse of language, is called a Grassman variable.
The Grassman variables are the basis vectors of a vector space (of dimension n) They form an algebra over a field, with the field usually being taken to be the complex numbers, although one could contemplate other fields, such as the reals. The algebra is a unital algebra, and the generators are anti-commuting:
Since the form a vector space over the complex numbers, it is trivial that they commute with the complex numbers; this is by defintion. That is, for complex x, one has
The squares of the generators vanish:
In other words, a Grassmann variable is a non-zero square-root of zero.
Let V denote this n-dimensional vector space of Grassman variables. Note that it is independent of the choice of basis. The corresponding exterior algebra is defined as
where is the exterior product and is the direct sum. The individual elements of this algebra are then called Grassman numbers. It is standard to completely omit the wedge symbol when writing a Grassman number; it is used here only to clearly illustrate how the exterior algebra is built up out of the Grassman variables. Thus, a completely general Grassman number can be written as
where the c's are complex numbers, or, equivalently, is a complex-valued, completely antisymmetric tensor of rank k. Again, the can be clearly seen here to be playing the role of a basis vector of a vector space.
Observe that the Grassmann algebra generated by n linearly independent Grassmann variables has dimension 2n; this follows from the binomial theorem applied to the above sum, and the fact that the n+1-fold product of variables must vanish, by the anti-commutatino relations, above. In most physics applications, the limit is assumed, and thus the sum above extends to infinity as well.
Any product of an odd number of Grassman variables is anti-commuting; such a product is often called an a-number. Any product of an even number of Grassman variables is commuting, and is often called a c-number. By abuse of terminology, an a-number is sometimes called an anticommuting c-number. This decomposition into even and odd subspaces provides a grading on the algebra; thus Grassmann algebras are the prototypical examples of supercommutative algebras. Note that the c-numbers form a subalgebra of , but the a-numbers do not (they are a subspace, not a subalgebra).
The definition of Grassman numbers allows mathematical analysis to be performed, in analogy to analysis on complex numbers. That is, one may define superholomorphic functions, define derivatives, as well as defining integrals.
Integrals over Grassman numbers are known as Berezin integrals. In order to reproduce the path integral for a Fermi field, the definition of Grassmann integration needs to have the following properties:
- partial integration formula
This results in the following rules for the integration of a Grassmann quantity:
Thus we conclude that the operations of integration and differentiation of a Grassmann number are identical.
number with A being an N × N matrix.
Grassmann numbers can be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers and . These Grassmann numbers can be represented by 4×4 matrices:
In general, a Grassmann algebra on n generators can be represented by 2n × 2n square matrices. Physically, these matrices can be thought of as raising operators acting on a Hilbert space of n identical fermions in the occupation number basis. Since the occupation number for each fermion is 0 or 1, there are 2n possible basis states. Mathematically, these matrices can be interpreted as the linear operators corresponding to left exterior multiplication on the Grassmann algebra itself.
In quantum field theory, Grassmann numbers are the "classical analogues" of anticommuting operators. They are used to define the path integrals of fermionic fields. To this end it is necessary to define integrals over Grassmann variables, known as Berezin integrals.
Grassmann numbers are also important for the definition of supermanifolds (or superspace) where they serve as "anticommuting coordinates". In the 1970s a completely new type of symmetry was discovered which provided a physical mechanism to cancel infinite ground state fluctuations. Supersymmetry is a trait of contemporary mathematical models that can be described in several ways. We can say first of all that spacetime has dimensions other than those to which we daily experience: they are called the Grassmann dimensions because they are measured with Grassmann numbers rather than ordinary real numbers. This formalism can be used to very efficiently write down supersymmetric quantum field theories.
There are some generalisations to Grassmann numbers. These require rules in terms of N variables such that:
where the indices are summed over all permutations so that as a consequence:
for some N > 2. These are useful for calculating hyperdeterminants of N-tensors where N > 2 and also for calculating discriminants of polynomials for powers larger than 2. There is also the limiting case as N tends to infinity in which case one can define analytic functions on the numbers. For example, in the case with N = 3 a single grassmann number can be represented by the matrix:
so that . For two grassmann numbers the matrix would be of size 10×10.
For example, the rules for N = 3 with two Grassmann variables imply:
so that it can be shown that
which gives a definition for the hyperdeterminant of a 2×2×2 tensor as