# I have never seen a Grassmann Number

1. May 25, 2012

### the_pulp

Is there a way to represent Grassmann Numbers from previously known mathematical entities? Something like when it is said, for "C", that z=x+i*y and i^2=-1 or that z = [a -b; b a] with the usual rules of matrix sum and multiplication?

It is pretty strange to me that it is so hard to find books or pdfs on line about it. has someone ever demonstrated that these entities even exist? Is fermionic math supported over nothing????

2. May 25, 2012

### Dickfore

Have you ever thought about asking yourself why they are called 'Grassmann' numbers in the first place?

3. May 25, 2012

### the_pulp

Yes, for 1 second, and I found that there was a guy called grassmann that invented this object. Why are you asking? (Im perceiving sarcasm)

Just to let you know, before posting, I have been reading wikipedia and some books, and there is something called Grassmann Algebra and wedge product, which seems to be the mathematical construction Im looking for. But this wedge product is not exactly anticommuting, In fact it is conmuting or anticommuting depending on the grade of the elements of the Grassmann Algebra we are "wedge multiplying". So, is there any set of objects that, independently of which pair of objects we use, the product anticonmutes?

Thanks!

4. May 25, 2012

### The_Duck

I think it's valid to view Grassmann numbers as simply a convenient mathematical trick for encoding the rules for calculating certain things in QFT. Observables are always real numbers, not Grassmann numbers. Grassmann numbers "exist" in the sense that they can be consistently defined as abstract objects satisfying a certain set of rules. Once you've defined what you mean by an integral over a Grassmann variable, it's possible to represent the rules for calculating observables for processes involving fermions in terms of a path integral over Grassmann-valued fields. But this is just a formal technique for computing some real number.

5. May 25, 2012

### Hurkyl

Staff Emeritus
That (plus some other details) is just the definition of the technical term. This fact doesn't have any significance beyond technical convenience, and it's roughly analogous to the fact it's sometimes convenient to compute/manipulate a complex number in terms of its real and imaginary parts, and to a lesser extent analogous to the fact it's occasionally convenient to compute/manipulate vectors and linear transformations in terms of their coordinates with respect to a basis.

6. May 26, 2012

### fzero

The Grassmann numbers are precisely the objects that anticommute amongst themselves. Their introduction is analogous to introducing $i$ satisfying $i^2=-1$ to generate complex numbers. We add abstract objects that satisfy $\theta_i\theta_j = - \theta_j\theta_i$ to ordinary vector spaces to obtain Grassmann algebras.

At http://en.wikipedia.org/wiki/Grassmann_number#Matrix_representations a representation of Grassmann numbers using matrices is given, but this is rarely used in physics examples.

7. Jun 4, 2012

### A. Neumaier

Precisely the same happens for Grassmann numbers. If a, b, c anticommute with each other then, necessarily, ab, ac, and bc commute with a,b,c.

More generally, in an arbitrary associative algebra, the product of an even number of mutually anticommuting elements commutes with everything, whereas the products an odd number of mutually anticommuting elements anticommute with each other (but not with the even products). This is why Grassmann numbers form a so-called graded algebra, with complementary even and odd subspaces, such that odd elements anticommute with each other, whereas even elements commute with everything.