Spinors in d dimensions and Clifford algebra

[nrqed]

I bought a book on susy and there is a chapter on spinors in d-dimensions.
Now, maybe I am extremely dumb but I just can't understand the first few lines!

EDIT: I was being very dumb except that I think there is a typo...See below...


BEGINNING OF QUOTE

Consider a d-dimensional vector space V over the field F for which we shall choose two alternatives F=R or F=C. Let Q be a quadratic form on V:

$$Q:x \in V \rightarrow Q(x) \in F$$

This defines a symmetric scalar product

$$\phi(x,y) \equiv xy + yx = Q(x+y) - Q(x) - Q(y)$$

In particular, for $$e_\mu , \mu = 1 \ldots d,$$ a basis of V, orthogonal with respect to Q, we then have

$$e_\mu e_\nu + e_\nu e_\mu = 2 \delta_{\mu \nu} Q(e_\mu) \cdot 1 ~~~(3.1)$$

The associative algebra with unit element generated by the $e_\mu$ with the defining relation (3.1) is called the Clifford algebra C(Q) of the quadratic form Q. The dimension of C(Q) is $2^d$


END OF QUOTE

Questions:



1) What does it mean to say that the basis is orthogonal with respect to Q? Does that mean $$Q(e_\mu + e_\nu) = 0$$ unless mu = nu? Or is that a typo an dhe meant to write orthogonal with respect to $$\phi$$, not Q?

2) No matter what I try I don't see how to get from the definition of phi to equation 3.1!


EDIT: I THINK I GOT IT

Just as I posted I think that I finally understood.

First, I think the author really meant that the basis is orthogonal with respect to phi and not Q. In that case, we get

$$\delta_{\mu \nu} (Q(2 e_\mu) - Q(e_\mu) - Q(e_\mu))$$

since the form is quadratic, this gives

$$\delta_{\mu \nu} (4 Q( e_\mu) - Q(e_\mu) - Q(e_\mu)) = 2 \delta_{\mu \nu} Q(e_\mu)$$


For some reason it just clicked after I posted my question!

 

[ophase]

I think i like the book...Who is the author of that book?? How can i buy it too??

 

[sir-pinski]

Sorry for the confusion.

RESPONSE:

No need to apologize, understanding new concepts can be challenging at times. Let me try to clarify the content on spinors in d dimensions and Clifford algebra for you.

First, let's define some terms. A spinor is a mathematical object used in physics and mathematics to describe the intrinsic angular momentum of a particle. It is represented by a mathematical symbol that has both magnitude and direction, similar to a vector. In d dimensions, we are working with a vector space that has d dimensions.

Now, let's focus on the quadratic form Q. This is a function that maps a vector x in the vector space V to a scalar value in the field F. This quadratic form is used to define a symmetric scalar product, denoted by the symbol phi. This product takes two vectors x and y and returns a scalar value, which is the sum of the products of their components.

Next, we have the basis e_mu, where mu is a number from 1 to d. This basis is chosen to be orthogonal with respect to the scalar product phi. This means that the scalar product of any two basis vectors is equal to 0 unless mu = nu. In other words, the basis vectors are perpendicular to each other.

Now, let's look at equation 3.1. It is derived from the defining relation (3.1) and the properties of the quadratic form. The quadratic form Q is used to evaluate the scalar product of the sum of two basis vectors, e_mu and e_nu. This gives us the expression Q(e_mu + e_nu). Using the properties of the quadratic form, we can expand this expression to get the equation 3.1.

I hope this helps clarify the content on spinors in d dimensions and Clifford algebra for you. Keep asking questions and exploring the topics, it will help you understand them better.