Geometry and Physics (Part 1)

Geometric Calculus Based on Clifford Algebra

In 1992 I met prof. Waldyr A. Rodrigues, jr., who introduced me into the subject of Clifford algebras. We were guests of prof. Erasmo Recami at the Institute of Theoretical Physics, Catania, Italy. Until then, I used tensor calculus of general relativity, but Waldyr opened my eyes and showed me that tensor calculus, although very elegant and practical, has its limitations. Moreover, Clifford algebras are not only a useful tool for description of the existing physics and geometry,  but they can also be used for formulation of new physical theories. In this  series of posts I would like to introduce the subject, and forward my enthusiasm with Clifford algebras to those readers, who are not yet fascinated by them. To the beginners  I recommend the books by D. Hestenes [1]

I am now going to discuss the calculus with vectors and their generalizations. Geometrically, a vector is an oriented line element.

How to multiply vectors? There are two possibilities:

1. The inner product

(1)              a \cdot b = b \cdot a

of vectors a and b. The quantity a · b is a scalar.

2. The outer product

(2)              a \wedge b = -b \wedge a

which is an oriented element of a plane. The outer product is the wedge product of two vectors, and is  called bivector. The above two products are the symmetric and antisymmetric part of the Clifford product, also called the geometric product:

(3)              a b = a \cdot b + a \wedge b


(4)              a \cdot b \equiv \frac{1}{2} (a b + b a)

(5)              a \wedge b \equiv \frac{1}{2} (a b - b a).

For an orthonormal set of vectors, e_i, ~,e_j,~~i,j =1,2,...,n, that span a vector space V_n,  we have the relations:

(6)           e_{i} \cdot e_{j} \equiv \frac{1}{2} (e_{i} e_ j + e_{j} e_{i}) = \delta_{i j}.

This is the defining relation of the Clifford algebra Cl(n)\,. The vector space V_n can be V_3, which is isomorphic to our three dimensional space that we live in.

We see that vectors of an n-dimensional space are Clifford numbers. Within Clifford algebra, calculus with vectors can be straightforwardly  performed, and extended to the calculus with bivectors, trivectors, etc., also called 2-vectors, 3-vectors, etc. , in general r-vectors:

In a space of finite dimension this cannot continue indefinitely: the n-vector is the highest r-vector in V_n and the (n+1)-vector is identically zero. An r-vector A_r represents an oriented r-volume  in V_n.

Multivectors A_r are elements of Clifford algebra Cl(n) of V_n. An element of Cl(n) will be called Clifford number. Clifford numbers can be multiplied among themselves and the results are Clifford numbers of mixed degrees, as indicated in the basic equation (3). The theory of multivectors, based on Clifford algebra, was developed by Hestenes [1]. In the following  some useful formulas are displayed without proofs.

For a vector a and an r-vector A_r, the inner and the outer product are defined according to

(7)           a \cdot A_r \equiv \frac{1}{2} \left ( a A_r - (-1)^r A_r a \right ) =    - (-1)^r A_r \cdot a

(8)             a \wedge A_r \equiv \frac{1}{2} \left ( a A_r + (-1)^r A_r a \right ) =    (-1)^r A_r \cdot a

The inner product has symmetry opposite to that of the outer product, therefore the signs in front of the second terms in the above equations are different.

Combining (7) and (8) we find

(9)           a A_r = a \cdot A_r + a \wedge A_r

For A_r = a_1 \wedge a_2 \wedge ... \wedge a_r\;\; eq.(7) can be evaluated to give the useful expansion

(10)         a \cdot (a_1 \wedge ... \wedge a_r) =\\ \sum_{k=1}^r (-1)^{k+1}(a \cdot (a_k) a_1 \wedge ... a_{k-1} \wedge a_{k+1} \wedge ... a_r)

In particular,

(11)       a \cdot (b \wedge c) = (a \cdot b)c - (a \cdot c) b

Let e_1, \, e_2, \, ..., \, e_n be linearly independent vectors, and  \alpha,\;\, \alpha^i\alpha^{i_1 i_2} scalar coefficients. A generic Clifford number can then be written as

(12)  \displaystyle A=\alpha +\alpha^i e_i +\frac{1}{2!}\alpha^{i_1 i_2} e_{i_1}\wedge e_{i_2} + ...\frac{1}{n!}\alpha^{i_1 ... i_n} e_{i_1}\wedge ...\wedge e_{i_n}

Since it is a superposition   of multivectors of all possible grades, it will be called polyvector.  Following a suggestion by W. Pezzaglia, I call a generic Clifford number polyvector, and reserve the name  multivector for an r-vector, since the latter name is already widely used for the corresponding object in the calculus of differential forms. Another name, also often used in the literature, is Clifford aggregate. These mathematical objects have far reaching geometrical and physical implications that I will discuss and explore during the course of this blog.

To demonstrate the usefulness of Clifford algebras I give below some excerpts from my paper Found. Phys. 31 (2001) 1185  [arXiv:hep-th/0011216]
Algebra of Spacetime
Polyvector Fields

Physical Quantities as Polyvectors

The compact equations in the above excerpts suggest a generalization that every physical quantity is a polyvector. In this blog we shall explore such an assumption and see how far we can get.

In 4-dimensional spacetime the momentum polyvector is

(13)      P = \mu + p^{\mu} \gamma_{\mu} + S^{\mu \nu} \gamma_{\mu} \gamma_{\nu} +  \pi^{\mu} \gamma_5 \gamma_{\mu} + m \gamma_5 ,

and the velocity polyvector  is

(14)      {\dot X} = {\dot \sigma} + {\dot x}^{\mu} \gamma_{\mu} + {\dot \alpha}^{\mu \nu} \gamma_{\mu} \gamma_{\nu} + {\dot \xi}^{\mu} \gamma_5 \gamma_{\mu} + {\dot s} \gamma_5

where \gamma_{\mu} are four basis vectors satisfying

(15)     \gamma_{\mu} \cdot \gamma_{\nu} = \eta_{\mu \nu}

and \gamma_5 \ \equiv \gamma_0 \gamma_1 \gamma_2 \gamma_3 is the pseudoscalar.

We associate with each particle the velocity polyvector {\dot X} and the momentum polyvector P. These quantities are  generalizations of the point particle 4-velocity {\dot x} and its momentum p. Besides a vector part we now include the scalar part {\dot \sigma}, the bivector part {\dot \alpha}^{\mu \nu} \gamma_{\mu} \gamma_{\nu}, the pseudovector part {\dot \xi}^{\mu} \gamma_5 \gamma_{\mu} and the pseudoscalar part {\dot s} \gamma_5 into the definition of particle’s velocity, and analogously for particle’s momentum.

 [1] D. Hestenes, Space-Time Algebra (Gordon and Breach, New York, 1966);
D. Hestenes,  Clifford Algebra to Geometric Calculus (D. Reidel, Dordrecht, 1984)


About Matej Pavšič

I am theoretical physicist at Jožef Stefan Institiute
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One Response to Geometry and Physics (Part 1)

  1. amarashiki says:

    Dear Matej! I am happy to see you as a blogger! 🙂 I have delayed the post on your system units (I suppose you have seen it) due to I am organizing a full thread with every system of “natural units” I have ever seen in the literature. Indeed, your units will be the LOG#070 from a series of 5 or 6 additional posts. I was planning to introduce C-space relativity in my blog, but if you do it here, I could focus on some other aspects of that extension of relativity. Welcome to the blogsphere and good luck with your blog! I am sure it will be as interesting as your papers.
    Cheers and all the best!

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