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.

GeomSpinors07We 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,   [ ]

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) [ ]


About Matej Pavšič

I am theoretical physicist at Jožef Stefan Institiute
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2 Responses to From vectors to spinors, and beyond

  1. amarashiki says:

    Dear Matej. Excellent review or your ideas and your talk! Let me add two extra authors wondering about mirror matter: 1) Silagadze has also mentioned it here and he has talked about how useful it could be as a energy resource (almost as useful as a superheavy neutrino N_R or a heavy sterile N) and 2) Moreover, it is also a claim as well in the works of the ternary stuff by G.Volkov, like and references therein. It seems we need something for DM and DE, but I bet DM could be heavy neutrinos AND (quantum) gravitons. Dark Energy, I have no idea of what it could be being repulsive, but maybe it could also be some kind of illusion due to our current equations to model that “effect”. I am really wishing to see PLANCK data! If I am right, no DM experiment will confirm a detection of DM particles the next year (of course, I can be wrong!). Best wishes, Matej!

  2. amarashiki says:

    PS: and Happy New Year 2013! It is going to be an interesting year for Science.

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