From vectors to spinors, and beyond

Spinors are usually considered as weird objects that cannot be easily understood in intuitive terms. But within the context of  Clifford algebra Cl(1,3) there exist a well-known procedure to construct spinors in terms of vectors, and their wedge products. In the series of the following slides, taken from my talk at the 7th Mathematical Physics Meeting, September, 2012, Belgrade, Serbia, I show, how this can be done.

To enlarge click on picture

A Clifford number $\Phi$ in general transforms from the left and from the right. Let a transformation from the left be denoted  R, and a transformation from the right, S.

The above examples show that Clifford numbers, transformed from the left by R and from the right by S, in general change their grade decomposition. In particular, a vector can be transformed into a trivector, a scalar into a bivector, etc. Transformations of the form

$X' = {\rm R} X {\rm S}$

thus rotate within the Clifford algebra.

Now I will show how spinors arise within the Clifford algebra of Minkowski space. Instead of a vector basis $\gamma_0, \gamma_1, \gamma_2, \gamma_3$, we will take another basis, called the Witt basis.

The new basis vectors satisfy the fermionic anticommutation relations. They behave as creation and annihilation operators that act on the vacuum state $f$, and form a Fock space basis for spinors.

We see that spinors are elements of the Fock space spanned by the basis

$f, \; \theta_1 f, \; \theta_2 f, \; \theta_1 \theta_2 f$

There are four different ways to form a vacuum from the Witt basis vectors, and there are four different sets of creation and annihilation operators.

We have thus four different spinor spaces, whose direct sum is the Clifford algebra Cl(1,3).

We distinguish the passive and the active transformations.  In the case of a passive transformation, the object, e.g., a Clifford number $\Phi$, remains unchanged, whereas its components, $\psi^{\tilde A}$, and the basis elements, $\xi_{\tilde A}$,  do change.

Under Lorentz transformations, the basis vectors $\gamma_\mu$, $\mu=0,1,2,3$ transform in the well known way, from the left by ${\rm R}$, and from the right by ${\rm R}^{-1}$. This fact has led F. Piazzese [Clifford Algebras and their Applications in Mathematical Physics, F. Bracks et al. (eds.), 325-332  (Kluwer, 1993)] to the conclusion that spinors cannot be embedded in a Clifford algebra, because spinors transform only from the left.

But we have seen that the above transformations, $\Phi '= {\rm R} \Phi {\rm R}^{-1}$,  are not the most general transformations that act on Clifford numbers. The most general transformations are of the form $\Phi '= {\rm R} \Phi {\rm S}$. In particular it can be  ${\rm S} = 1$. Then the Clifford number transforms from the left only, just as a spinor.

We will now consider the behavior of Clifford numbers under the usual proper and improper Lorentz transformations.

Below we illustrate the case in which an observer, associated with a reference frame, spanned by the basis vectors $\gamma_1$, $\gamma_2, \gamma_3$,  starts to rotate and performs the “full” $2 \pi$ turn. The observer and his reference frame performs the turn, whereas the physical object, e.g.,  an electron described by the spinor $\Psi$ remains fixed. Under such a $2 \pi$ turn, every Clifford number, including the spinors $\Psi$, remains unchanged. No change of sign occurs in this example.

This was a passive transformations. There must also exist the corresponding active transformation, so that the observer and the reference frame remain fixed, whereas the object (spinors) performs a turn.

The well known transformation of a spinors, namely

$\Psi' = {\rm R} \Psi$ ,

is another kind of  transformation within the Clifford algebra, not the one considered in the above slide. As we have seen at the beginning of this post, such transformation can act on any Clifford number, not only on a spinor. It can act on a vector as well. If it acts on a vectors, then in general it transforms its grade, e.g., a vector into a trivector, or a vector into a scalar, etc.

I will now show  how a spinor transforms under a rotation and under the space inversion.

Under such rotation, a left handed spinor of the first ideal transforms into a left handed spinor of the second ideal. For more details about those rotations see the slides 19-21 of the talk presented at  7th Mathematical Physics Meeting, September, 2012, Belgrade, Serbia, where examples of the eigenstates of the spin operator $S_3=-\frac{i}{2} \gamma_1 \gamma_2$ are shown.  There exist a family of the eigenstates of $S_3$ with the same eigenvalues, the sates within the family being related to each other by an SU(2) transformation that mixes the first and the second ideal.

If we perform the space inversion, then the time-like vector $\gamma_0$ remains the same, whereas the space-like vectors $\gamma_1$, $\gamma_2$, $\gamma_3$ change the sign. As a consequence, the Witt basis vectors and spinors transform as shown below.

Under the space inversion, a left handed spinor of the first ideal transforms into a right handed spinor of the third ideal.

Generalized Dirac equation

If we consider a  physical situation in the mirror (which performs the space inversion apart from a rotation),  then a process becomes a mirror process. If the particles involved are weakly interacting fermions, then according to the setup considered above, they are described by the generalized spinors that can be represented as $4 \times 4$ matrices. Let us assume that such generalized spinors satisfy the generalized Dirac equation.

I call this equation the Dirac-Kahler equation, because it is closely related to the equation known in the literature under this name. From the construction given above, it is clear that the fields $\psi^{\alpha i}$ can be complex valued. The scalar product

$\langle (\xi^{\tilde A})^\ddag \gamma^\mu \xi_{\tilde B} \rangle_S \equiv {(\gamma^\mu)^{\tilde A}}_{\tilde B}$

is calculated by taking the scalar part of the Clifford product. In general, the scalar product of two or several complex valued Clifford numbers is defined here so that the scalar and the real part of the Clifford product is taken.  In our construction, contrary to that discussed in the literature, we are not confined to Majorana spinors. The above equation can describe four Dirac spinors as well.

Gauge invariant action

The gauge invariant equation contains the gauge field ${{G\mu}^i}_j$, where the indices i, j denote the four minimal left ideals of Cl(1,3) (i.e., the four columns).

The above construction contains the gauge group SU(2),  acting on the matrix $\psi^{\alpha i}$ from the right, and thus mixing the first and the second minimal left ideal (first and second column). This SU(2) can be interpreted as the weak interaction gauge group. The states of the first and second ideal (column) can be interpreted as the ordinary particles, whereas the states of the third and the fourth ideal (column) can be represented as mirror particles. The latter particles are obtained from the former ones by the space inversion.

Our generalized Dirac equation  thus includes the weak interaction. The transformation properties of the generalized spinors  explain the enigmatic behavior of the weak interaction processes under the space inversion. Under the space inversion, the ordinary particles interacting with the ordinary SU(2) weak interaction are transformed into the mirror particles, interacting with the mirror SU(2) weak interaction.

Mirror particles were first proposed by  Lee and Yang,   Phys. Rev. 104 (1956) 254. Subsequently, the idea of mirror particles and the exact parity model has been pursued by

I.Yu. Kobzarev, L.B. Okun, I.Ya. Pomeranchuk, Soviet J. Nucl. Phys. 5 (1966) 837.
M. Pavšič, Int. J. Theor. Phys. 9 (1974) 229.
E.W. Kolb, D. Seckel, M.S. Turner, Nature 314 (1985) 415
R. Foot, H. Lew, R.R. Volkas, Phys. Lett. B 272 (1991) 67;
R. Foot, H. Lew, R.R. Volkas, Mod. Phys. Lett. A 7 (1992) 2567;
R. Foot, Mod. Phys. Lett. 9 (1994) 169;
R. Foot, R.R. Volkas, Phys. Rev. D 52 (1995) 6595.

The possibility that mirror particles are responsible for dark matter has been explored in many works, e.g.:

H. M. Hodges, Phys. Rev. D 47 (1993) 456;
R. Foot, Phys. Lett. B 452 (1999) 83;
R. Foot, Phys. Lett. B 471 (1999) 191;
R.N. Mohapatra, Phys. Rev. D 62 (2000) 063506;Z. Berezhiani, D. Comelli, F. Villante, Phys. Lett. B 503 (2001).
P. Ciarcelluti, Int. J. Mod. Phys.D14 (2005) 187;
P. Ciarcelluti, Int. J. Mod. Phys.D14 (2005) 223;
P. Ciarcelluti, R. Foot, Phys. Lett. B679 (2009) 278.

A demonstration that mirror particles can be explained in terms of the algebraic spinors (elements of Clifford algebras) was presented in

M. Pavšič, Phys. Lett. B 692 (2010) 212, http://dx.doi.org/10.1016/j.physletb.2010.07.041   [ http://arxiv.org/abs/arXiv:1005.1500 ]

In order to obtain other interactions, one has to generalize the model discussed above. I will say more about this in one of my next posts. For the time being the reader may consult the paper  “A Novel View on the Physical Origin of E8″,  Journal of Physics A: Mathematical and Theoretical  41, 332001 (2008) http://dx.doi.org/10.1088/1751-8113/41/33/332001 [ http://arxiv.org/abs/arXiv:0806.4365 ]

Posted in Uncategorized | | 2 Comments

Geometry and Physics (Part 2)

Clifford Space: An Extension of Spacetime

An extended object, O, can be be sampled by a finite set of parameters, for instance, by the center of mass coordinates, and by the orientation of its axes of symmetries. Higher multipole deformations, such as the dipole and the quadrupole ones, can also be taken into account. For practical reasons, only a finite number of multipoles can be taken into account. Instead of the infinite number of degrees of freedom, we consider only a finite number of degrees of freedom. We thus perform a mapping from an infinite dimensional configuration space, associated with the object O, to a finite dimensional subspace.

Extended objects of particular interest for theoretical physics are strings and branes. They can be described by coordinate functions $X^\mu (\xi^a)$, $\mu=0,1,2,...,N-1$, $a=0,1,2,...,n-1$, where $n\le N$. Such a description is infinite dimensional. In refs. [1] it was pointed out how one can employ a finite description in terms of a quenched mini superspace.

The idea has been further developed [2]–[6] by means of Clifford algebras, a very useful tool for description of geometry [7].

Click on pictures to enlarge them.

Here we are interested in description of spacetime, $M_N$, and the objects embedded in $M_N$.Therefore, let us start by considering the squared line element in $M_N$:

(1)        $Q = d s^2 = g_{\mu \nu} d x^\mu d x^\nu, \quad \quad \mu,\nu=0,1,2,...N-1$.

If we take the square root, $\sqrt{Q}$, we have the following possibilities:

i) $\sqrt{Q} = \sqrt{g_{\mu \nu} {\rm d} x^\mu d x^\nu}\;\;\;$ scalar

ii)  $\sqrt{Q} = \gamma_\mu d x^\mu\;\;\;$   vector

Here $\gamma_\mu$ are generators of the Clifford algebra $Cl(p,q)$,
p+q=N,  satisfying

(2)        $\gamma_\mu \cdot \gamma_\nu \equiv \frac{1}{2}(\gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu ) = g_{\mu \nu}$ ,

where $g_{\mu \nu}$ is the metric of $M_N$.

The generators $\gamma_\mu$ have the role of basis vectors of the spacetime $M_N$. The symmetric product $\gamma_\mu \cdot \gamma_\nu$ represents the inner product. The antysymmetric (wedge) product of two basis vectors gives a unit bivector:

(3)        $\gamma_\mu \wedge \gamma_\nu \equiv \frac{1}{2}(\gamma_\mu \gamma_\nu - \gamma_\nu \gamma_\mu )$

and has thus the role of outer product. In analogous way we obtain 3-vectors, 4-vectors, etc..

We assume that the signature of an $N$-dimensional spacetime is $(1,N-1)$, i.e., $(+ - - - ...)$. In the case of the 4-dimensional spacetime we thus have the signature $(1,3)$, i.e., $(+ - - -)$. The corresponding Clifford algebra is $Cl(1,3)$.

The basis of $Cl(1,N-1)$ is

(4)        $\lbrace 1, \gamma_\mu, \gamma_{\mu_1} \wedge \gamma_{\mu_2},..., \gamma_{\mu_1} \wedge \gamma_{\mu_2} \wedge ... \wedge\gamma_{\mu_N}\rbrace$

A generic element, $X \in Cl(1,N-1)$, is a superposition

(5)        $X=\sum_{r=0}^N \frac{1}{r!} X^{\mu_1 \mu_2 ...\mu_r} \gamma_{\mu_1} \wedge \gamma_{\mu_2}\wedge ... \wedge\gamma_{\mu_r} \equiv X^M \gamma_M$

called a Clifford aggregate or polyvector.

In refs. [5,6,8] it has been demonstrated that r-vectors $X^{\mu_1 \mu_2 ...\mu_r}$ can be associated with closed instantonic (r-1)-branes or open instantonic r-branes. A generic polyvector, $X=X^M \gamma_M$, can be associated with a conglomerate of (instantonic) r-branes for various values of $r=0,1,2,...,N$.

Our objects are instantonic r-branes, which means that they are localized in spacetime*. They generalize  the concept of `event’, a spacetime point, $x^\mu,~\mu=0,1,2,3$. Instead of an event, we have now an extended event, ${\cal E}$,  described by coordinates $X^{\mu_1 \mu_2 ...\mu_r}$, $r=0,1,2,3,4$. The space of extended events is called Clifford space, $C$. It is a manifold whose tangent space at any of its points is a Clifford algebra $Cl(1,3)$. If $C$ is a flat space, then it is isomorphic to the Clifford algebra $Cl(1,3)$ with elements

(6)        $X=\sum_{r=0}^4 \frac{1}{4!} X^{\mu_1 \mu_2 ...\mu_r} \gamma_{\mu_1 \mu_2 ...\mu_r}\equiv X^M \gamma_M$.

In flat $C$-space, the basis vectors are equal to the wedge product

(7)        $\gamma_M = \gamma_{\mu_1} \wedge \gamma_{\mu_2}\wedge ... \wedge \gamma_{\mu_r}$

at every point ${\cal E} \in C$. This not true in curved $C$-space: if  we (parallelly) transport a polyvector $A=A^M \gamma_M$from a point ${\cal E} \in C$ along a closed path back to the original point, ${\cal E}$, then the orientation of the polyvector $A$ after such transport will not coincide with the initial orientation of $A$. After the transport along a closed path we will obtain a new polyvector $A'=A'^M \gamma_M$. If,in particular, the initial polyvector is one of the Clifford algebra basis elements, $A=\gamma_M$, i.e., an object with definite grade, then the final polyvector will be $A'=A'^M \gamma_M$, which is an object with mixed grade. A consequence is that in curved Clifford space $C$, basis vectors cannot have definite grade at all points of $C$.

The situation in a curved Clifford space, C,  is analogous to that in a usual curved space, where after the (parallel) transport along a closed path, a vector changes its orientation. In Clifford space, a change of orientation in general implies a change of a polyvector’s grade, so that, e.g., a definite grade polyvector changes into a mixed grade polyvector.

However, if we impose a condition that, under parallel transport, the grade of a polyvector does not change, then one has a very special kind of curved Clifford space [9]. In such a space, afer a parallel transport along a closed path, the vector part $\langle A \rangle_1 = a^\mu \gamma_\mu$ changes into $\langle A' \rangle_1 =a'^\mu \gamma_\mu$, the bivector part $\langle A \rangle_2 =a^{\mu \nu} \gamma_\mu\wedge \gamma_\nu$ changes into $\langle A' \rangle_2 =a'^{\mu \nu} \gamma_\mu \wedge \gamma_\nu$, etc., but one grade does not change into another grade. Such special Clifford space, in which the consequences of curvature manifest themselves within each of the subspaces with definite grade separately, but not between those subspaces, is very complicated. We will not consider such special Clifford spaces, because they are analogous to the usual curved spaces of the product form $M= M_1 \times M_2 \times...M_n$, where $M_i \subset M$ is a curved lower dimensional subspace of $M$, and where only those (parallel) transports are allowed that bring tangent vectors of $M_i$ into another tangent vectors of the same subspace $M_i$.

The squared line element in Clifford space, $C$, is

(8)        ${\rm d} S^2 = G_{M N} {\rm d} x^M x^N = {\rm d} X^\dagger {\rm d} X = \langle {\rm d} X^\ddagger {\rm d} X \rangle_0$ .

Here ${\rm d} X = {\rm d} x^M \gamma_M$, and ${\rm d} X^\ddagger = {\rm d} x^M \gamma_M^\ddagger$, where $\ddagger$ denotes the operation of inversion: $(\gamma_{\mu_1}\gamma_{\mu_2} ... \gamma_{\mu_r})^\ddagger = \gamma_{\mu_r}\gamma_{\mu_{r-1}} ...\gamma_{\mu_r}$. The metric of  C is

(9)        $G_{MN} = \gamma_M^\ddagger * \gamma_N = \langle \gamma^\ddagger \gamma_N \rangle_0$ ,

where $\langle ~~\rangle_0$ means the scalar part. A Clifford space with such a metric has signature [6] (8,8), i.e., $(++++++++--------)$. This is ultrahyperbolic space with neutral signature.

In the paper “Quantum Field Theories in Spaces with Neutral Signatures”[http://arxiv.org/abs/arXiv:1210.6820]  it is shown that, contrary to the wide spread belief, the physics in spaces with signature $(n,n)$ makes sense.

*The usual p-branes are localized in space, but they are infinitely extended into a time-like direction, so that they are (p+1)-dimensional worldsheets in spacetime.}

References

[1] Ansoldi S, Aurilia A, Castro C and Spallucci E 2001 Phys. Rev. D 64 026003 [arXiv:hep-th/0105027]; Aurilia A, Ansoldi S and Spallucci E 2002 Class.   Quant.    Grav. 19 3207  [arXiv:hep-th/0205028].
[2] Castro C 1999 Chaos, Solitons and Fractals 10 295 Chaos, Solitons and Fractals   12 (2001) 1585; Castro C and Pavšič M 2002; Phys. Lett. B 539 133 [arXiv:hep-th/0110079]; Castro C and Pavšič M 2005 Prog. Phys. 1 31
[3] Pavšič 2001 The Landscape of Theoretical Physics: A Global View; From Point       Particles to the Brane World and Beyond, in Search of a Unifying Principle (Dordrecht:   Kluwer)
[4] Pavšič M 2001 Found. Phys. 31 1185 [arXiv: hep-th/0011216].
[5] Pavšič M 2003 Found. Phys. 33 1277 [arXiv: gr-qc/0211085].
[6] Pavšič M 2007 Found. Phys. 37 1197 [arXiv: hep-th/0605126].
[7] Hestenes D 1966 Space-Time Algebra (New York:Gordon and Breach)
Hestenes D and Sobcyk G 1984 Cliff ord Algebra to Geometric Calculus (Dordrecht:
Reidel).
[8] Pavšič M 2012 Localized Propagating Tachyons in Extended Relativity Theories, arXiv: 1201.5755 [hep-th].
[9] Castro C 2012 Int. J. Theor. Phys. DOI 10.1007/s10773-012-1295-3
Castro C 2012 Adv. Appl. Cli ord Algebras DOI 10.1007/s00006-012-0370-4

Posted in Uncategorized | | 1 Comment

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)              $a b = a \cdot b + a \wedge b$

where

(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)

Posted in Uncategorized | | 1 Comment