- Grassmann number
In

mathematical physics , a**Grassmann number**(also called an**anticommuting number**or**anticommuting**) is a mathematical construction which allows a path integral representation for Fermionic fields. They were discovered byc-number David John Candlin in 1956 [*cite journal|journal= Nuovo Cimento | author= D.J. Candlin | volume =4 | title = On Sums over Trajectories for Systems With Fermi Statistics|pages=224 | year=1956*] . A collection of Grassman variable $heta\_i$ are independent elements of an algebra which contains the real numbers thatanticommute s with each other but commute with ordinary numbers $x$:: $heta\_i\; heta\_j\; =\; -\; heta\_j\; heta\_iqquad\; heta\_i\; x\; =\; x\; heta\_i.$

In particular, the square of the generators vanish:

: $(\; heta\_i)^2\; =\; 0,$, since $heta\_i\; heta\_i\; =\; -\; heta\_i\; heta\_i.$

In order to reproduce the path integral for a Fermi field, the definition of grassman integration needs to have the following properties:

* linearity: $int,\; [a\; f(\; heta)\; +\; b\; g(\; heta)\; ]\; ,\; d\; heta\; =\; a\; int,f(\; heta),\; d\; heta\; +\; b\; int,g(\; heta),\; d\; heta$

* partial integrations formula

: $int\; left\; [frac\{partial\}\{partial\; heta\}f(\; heta)\; ight]\; ,\; d\; heta\; =\; 0$

This results in the following rules for the integration of a Grassmann quantity:

: $int,\; 1,\; d\; heta\; =\; 0$: $int,\; heta,\; d\; heta\; =\; 1$

Thus we conclude that the operations of integration and differentiation of a Grassmann number are identical.

In the

path integral formulation ofquantum field theory the followingGaussian integral of Grassmann quantities is needed for fermionic anticommuting fields:: $int\; expleft\; [\; heta^TAeta\; ight]\; ,d\; heta,deta\; =\; det\; A$

with $A$ being a $N\; imes\; N$ matrix.

The algebra generated by a set of Grassmann numbers is known as a

Grassmann algebra . The Grassmann algebra generated by "n" linearly independent Grassmann numbers has dimension 2^{"n"}. These concepts are all named forHermann Grassmann .Grassmann algebras are the prototypical examples of

supercommutative algebra s. These are algebras with a decomposition into even and odd variables which satisfy a graded version ofcommutativity (in particular, odd elements anticommute).**Exterior algebra**The Grassmann algebra is the

exterior algebra of thevector space spanned by the generators. The exterior algebra is defined independent of a choice of basis.**Matrix representations**Grassmann numbers can always be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers $heta\_1$ and $heta\_2$. These Grassmann numbers can be represented by 4×4 matrices:

:$heta\_1\; =\; egin\{bmatrix\}0\; 0\; 0\; 0\backslash 1\; 0\; 0\; 0\backslash 0\; 0\; 0\; 0\backslash 0\; 0\; 1\; 0\backslash end\{bmatrix\}qquad\; heta\_2\; =\; egin\{bmatrix\}0000\backslash 0000\backslash 1000\backslash 0-100\backslash end\{bmatrix\}qquad\; heta\_1\; heta\_2\; =\; -\; heta\_2\; heta\_1\; =\; egin\{bmatrix\}0000\backslash 0000\backslash 0000\backslash 1000\backslash end\{bmatrix\}$

In general, a Grassmann algebra on "n" generators can be represented by 2

^{"n"}× 2^{"n"}square matrices. Physically, these matrices can be thought of asraising operator s acting on aHilbert space of "n" identicalfermion s in the occupation number basis. Since the occupation number for each fermion is 0 or 1, there are 2^{"n"}possible basis states. Mathematically, these matrices can be interpreted as the linear operators corresponding to left exterior multiplication on the Grassmann algebra itself.**Applications**In

quantum field theory , Grassmann numbers are the "classical analogues" of anticommutingoperators. They are used to define the path integrals offermionic field s. To this end it is necessary to define integrals over Grassmann variables, known asBerezin integral s.Grassmann numbers are also important for the definition of

supermanifold s (orsuperspace ) where they serve as "anticommuting coordinates".**References**

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