On Physical Significance of Spaces with Neutral Signature

Negative Energies in Classical Theories

We have seen that the Clifford space, C,  is a manifold, whose tangent space at any of its points is the Clifford algebra Cl(1,3). Its metric has the signature  (8,8), which means that it can be represented as the diagonal  matrix with eight times  +1 and eight times -1. Minkowski space has signature (1,3) and is an example of a hyperbolic space. In general, spaces with a signature, (p,q), p>1, are called ultra hyperbolic spaces. If p=q, the space is called to have neutral signature. The Clifford space C is an example of a space with neutral signature. Ultra hyperbolic spaces are generally believed to be unsuitable for physics, because they imply negative energies. I will demonstrate that negative energies, if properly treated, can make sense in physics. One known example is the Dirac theory of a relativistic particle, where the issue of negative energies was successfully resolved, although initially it presented a rather big puzzle [1]. Here I will talk about other cases, where negative energies are still considered as problematic. I will discuss a simple system in two dimensions. Generalization to higher dimensions and field theory is straightforward.

Harmonic oscillator in the space M_{1,1}

1. Free oscillator, without interaction
Let us consider the harmonic oscillator described by the Lagrangian

(1)        L =\frac{1}{2}\, (\dot x^2 -\dot y^2)  -\frac{1}{2}\, \omega^2 (x^2 - y^2) .

The corresponding equations of motion are

(2)        \ddot x + \omega x = 0,~~~~\ddot y + \omega y=0

We see that the change of sign in front of the y-term in the Lagrangian has no influence on the equations of motion.

The canonical momenta are

(3)         p_x = \frac{{\partial L}}{{\partial\dot x}} = \dot x\,~~~~p_y = \frac{{\partial L}}{{\partial \dot y}} = -\dot y

and the Hamiltonian is

(4)      H = p_x \dot x + p_y \dot y - L = \frac{1}{2}(p_x^2 - p_y^2) + \frac{{\omega^2}}{2}(x^2 - y^2).

It can be positive or negative, but this does not mean that the system is unstable. Namely, the Hamilton equation of motion are now

(5)         {\ddot x} = - \frac{\partial V}{\partial x}~,~~~~~~~  {\ddot y} = \frac{\partial V}{\partial y}~,~~~~  V=\frac{{\omega^2}}{2}(x^2 - y^2).

Notice that the y-equation has different sign in front of the force term than has the x-equation. This is the reason that also the motion of the y-component is stable, because it is described by the equation of motion of the form (2). The criterion for stability of the y-degree of freedom is that the potential must have a maximum in the (y,V)-plane [2-4]. We see that the usual intuition, namely that the system is stable if the potential has minimum, does not hold for the degrees of freedom with negative energy. For such degrees of freedom,  the system is stable if the potential has maximum.

Among certain experts [4-7] it is well known that the system, described by the Lagrangian (1), and other analogous, more general systems in field theory, are not problematic neither in the classical nor in the quantized theory, provided that there is no interaction between the degrees of freedom with positive and negative energy. But if one switches on an interaction, then, according to the wide spread belief, the system unavoidably becomes unstable so that  its position and velocity escape into infinity. We shall now demonstrate that this is not always the case, and that  even in the presence of interactions, the system with positive and negative energy degrees of freedom can be stable.

2. Presence of interactions

In general,  the interacting  oscillator in the space M_{1,1} is given by

(6)          L = \frac{1}{2}(\dot x^2 - \dot y^2) - V
(7)          V = \frac{\omega}{2}(x^2 - y^2) + V_1 .

Let us now consider the case of the interaction

(8)          V_1 = \frac{\lambda }{4}(x^2 - y^2)^2 .

The equations of motion are then

(9)          \ddot x + \omega ^2 x + \lambda \,x (x^2 - y^2) =0,
(10)         \ddot y + \omega ^2 y + \lambda \,y (x^2 - y^2 ) = 0.

These equations can be solved numerically by the program Mathematica. Let us take  \omega =1 and  \lambda = 0.1, and the initial conditions {\dot x} (0)=1, {\dot y}(0)=0, x(0)=0, y(0)=1. In the figure below it is shown how the trajectory in the (x,y) space starts to evolve in time. The time period in this figure is from t=0 to  t=25. The units are arbitrary.

ROscillator2Fig0

If we take a longer time period, e.g., from t=0 to t=100, then we obtain the following picture:

ROscillator1aFigcWe see that at t=100, the amplitudes of oscillations are  x\approx 10 and y\approx 10, whereas at t=25 the amplitudes were x=1.2 and y=1.2. Numerical calculations for larger values of t reveal that the system is unstable in the sense that the amplitudes of x(t), y(t){\dot x}(t), \dot y(t)  grow to infinity. In the figure bellow it is shown how the kinetic energy {\dot x}^2/2 increases with time.

Oscillator1aFig3c

Such results, of course, were expected, because the system described by the Lagrangian (6)-(8) has negative energies, and it is well known that negative energies imply instability. However, the total energy,

(11)         E_{\rm tot}=\frac{1}{2}({\dot x}^2 - {\dot y}^2)+V(x,y),

remains constant. For the above numerical solution, this is confirmed in the following figure, where the total energy remains constant within numerical error.

Oscillator1aFig4c
If we change the initial conditions or the coupling constant \lambda, then we obtain, of course,  different solutions that also escape into infinity. An interesting alternative to the (x,y) diagram above, that I do not show here,  can be found in Ref. [8].

Let us now consider slightly modified equations of motion by introducing two constants \mu and \nu:

(12)         \ddot x + \mu [\omega^2 x + \lambda \,x (x^2 - y^2)] = 0,

(13)         \ddot y + \nu [\omega^2 y + \lambda \,y (x^2 - y^2)] = 0.

The solution for \mu = 1.01,~\nu = 1, and the same initial conditions as before, is [8]:

Oscillator3  Left: trajectories in the (x,y). space.                                    Right: the kinetic energy  {\dot x}^2/2  as   function  of   time.

Now something really fascinating has happened: The system is no longer unstable! We see that the trajectory in the (x, y) space does not escape into infinity. Instead, a second arm is formed, and the trajectory remains confined within a star-like envelope. The kinetic energy of one component, {\dot x}^2/2, now does not grow to infinity. Instead, the amplitude of the kinetic energy oscillations is modulated by a slowly oscillating envelope. Rapid oscillations within the envelop are not visible in the diagram.

In the lower part of the latter figure  we repeat the calculation for \mu=1.0001 and \nu =1. This is now very close to \mu=1, \nu=1, the case of Eqs. (9),(10). Now the trajectory in the (x,y) space goes much farther from the origin than in the case \mu=1.01. But again it does not go to infinity; instead, after some time, the trajectory start to move within a second arm. The envelope of the kinetic energy oscillations now consists of separated peaks. As  \mu approaches 1, the height of the peaks becomes higher and higher, and their separation increases. In the limit \mu \to 1, the height of the first peak is infinite, and there is no second or other peaks (because their positions recedes to infinity). This is a singular case, in which the system is unstable. But if \mu slightly differs from \nu, then the system is stable; its trajectory remains confined within a finite region of the (x,y) space, and the kinetic energy remains finite as well.

3. Collision of the oscillator  with a free particle

Let us assume that in the surroundings of the oscillator, O, described by the Lagrangian (6)-(8), there is free particle, P. Depending on the initial conditions, it may happen that the oscillator hits the particle. Such a combined system of O and P can be modeled by the Lagrangian given in the next figure.

Click on figure to enlarge it
CollisionEq*See footnote

Let us solve this system for the constants \lambda =0.1~, \alpha=1, and the initial conditions \dot x(0) = 1, ~~\dot y(0) = 0, ~~\dot u(0) = 0, ~~\dot v(0) = 0, x(0) = 0, ~~y(0) = 1, ~~u(0) = 12, ~~v(0) = 11.5. The results are shown in this picture:

Collision2We see that the solution properly reproduces the fact that the particle P is initially at rest, and after the interaction with the oscillator O, it moves with a constant velocity. The total energy of the system remains constant, as shown in the lower left diagram of the above figure.

The positive component of the oscillator’s kinetic energy starts to increase, but at time around t=112 it drops down to zero, because the energy was transferred to the particle P. After a while, the oscillator “recovers”, and its positive energy starts to increase again. A further collision with some other particle would again drop down {\dot x}^2/2. Analogous holds for {\dot y}^2/2. We see that, according to this numerical solution, the surrounding particles stabilize the oscillator and prevent it to escape into the infinity. We expect that many such oscillators, immersed into a bath of particles would increase their average {\dot u}^2/2 and {\dot v}^2/2, i.e., the temperature of the bath. We see that this is a fascinating system, and it would be even more fascinating, if such systems could actually exist in nature (see a discussion at the end of Ref. [8]).

In the following posts I intend to say more on the fascinating world of negative energies. Among others, I will point out that negative energies also occur in the systems described by higher derivatives, and that such systems are not so problematic as it has been believed so far [9],[10].

  [1] E. Fermi 1932 Rev. Mod. Phys. 4 87.
[2] M. Pavšič 1999 Phys. Lett. A 254 119 [hep-th/9812123].
[3] M. Pavšič 2005 Found. Phys. 35 1617 [hep-th/0501222].
[4] R. Woodard 2007 Lect. Notes Phys. 720 403.
[5] W. Pauli 1943 Rev. Mod. Phys. 15 175.
[6] D. Cangemi, R.  Jackiw,  and B Zwiebach  1996 Annals of Physics 245  408.
[7] E. Benedict, R. Jackiw  and H.J. Lee 1996 Phys. Rev. D 54 6213
[8] M. Pavšič  J. Phys. Conf. Ser. 437  012006 (2013) arXiv:1210.6820 [hep-th].
[9] M. Pavšič 2013 Mod. Phys. Lett. A 28 1350165 arXiv:1302.5257 [gr-qc]
[10] M. Pavšič 2013 Phys. Rev. D 87  107502  arXiv:1304.1325 [gr-qc].

*In this figure we corrected the sign in front of the last term in the y-equation. The correct sign is minus, whereas in the initial version of this post the sign was plus. The wrong sign is also in Ref.[8], Eq. (89). This is a typo. The calculation was done with the correct minus sign in the y-equation.
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“The Trouble with Physics”

It is only recently that I had the opportunity to read the book The Trouble with Physics by Lee Smolin. I liked the book, and I was pleased to find that the author discussed some issues about fundamental theoretical physics that I have also been thinking about. Amongst others, Smolin analyzes in-depth why in the last two or three decades theoretical physics has not produced new results of fundamental significance. The current academic system  does not encourage young scientists with their own research projects. A young theoretical physicist must join an existing major project in order to have chances for a successful scientific career. That most young theoretical physicists join an existing research project is not bad in itself. A problem arises if nobody or very few ones with their own ideas can start working in an academic institution. Smolin observes that there are two types of scientists, and that science needs both: Seers and craftspeople. Seers are “good at asking genuinely novel but relevant questions, [ …], and have the ability to look at the state of a technical field and see a hidden assumption or a new avenue of research”. Unfortunately, the academic institutions do not embrace such creative rebels with this rare talent. Mostly they exclude them. Smolin observes that nowadays, there is place only for “craftspeople”, who usually master their discipline better than seers, but have no genuinely new ideas of their own. A consequence is that fundamental theoretical physics is no longer advancing as it has been during the last two centuries, when every twenty years or so there has been a major breakthrough discovery.

In the book Smolin lucidly discusses the sociology that has led to such a state of affairs. Let me add here my own opinion, or perhaps an observation, that  the progress was slowed down when people started to consider the number of citations, received shortly after publication, too seriously as a measure of the importance of an article . To have a chance in attracting soon many citations, a scientific paper has to be about a subject or a field that is being investigated by many researchers. It has to be about hot topics! The journals nowadays favor the publication of papers that are of “general scientific interest”, which in fact means that their subject must fit into one of the major fields that are being investigated at the time of submission of the paper. The policy is clear: such a paper is likely to attract citations soon, and thus contribute to the journal `impact factor’.  Nowadays, the journals by their policy exclude the papers written by “seers”, whose pioneering investigation is by definition not yet of “general scientific interest”, because  seers investigate topics that so far have not been investigated at all. The journals that compete for high impact factors are thus no longer serving to the development of fundamental theoretical physics in particular and to science in general as well as they could. A paper of “general scientific interest” can only be  an incremental advancement to what has already been investigated by many others. This is not bad in itself, bad is if nowadays  a “revolutionary” paper cannot be published at all. Therefore, as I perceive it, the invention of `impact factor’, and especially its usage as a measure of the importance of a journal, has led to such a “sociology” within the community of theoretical physicists that resulted in the stalled progress.

To point out that  “impact factor”  cannot be taken too seriously as a measure of a journal’s importance for the development of science, I am now asking: What was the impact factor of the not so well-known journal of Brno (German: Brünn)  in which Mendel had published his laws of genetics? His paper was appreciated only thirty five years later by renown scientists, who made Mendel’s work known to the scientific community. The “impact factor” (that takes into account citations within the two-year period only)  of that relatively obscure journal was–I guess– much lower than the impact factor  of the renown journals in Mendel’s time. However, given the fundamental importance and influence of Mendel’s paper for genetics, it would be absurd to say that the impact to science of that journal was as insignificant as suggested by its “impact factor”. Mendel published his paper in a relatively unknown journal, got very few citations in the first thirty five years, and  yet this turned out to be one of the greatest papers of all times.

The “impact factor” has yet another damaging consequence: Many people read only the journals with high impact factors and ignore those with low impact factors. However, nowadays an important “revolutionary” paper is very likely to be rejected by renown journals, and eventually appear in a journal with a low impact factor. But since those journals have not many readers, such a paper will remain unnoticed for long time. Fortunately, we have also preprint servers, such as arXiv. But scientists outside academic institutions cannot post their works there, unless they obtain an endorsement from an established scientist. So nowadays a “new Einstein” has a really hard time to be noticed at all. Of course, most would be “new Einsteins” are cranks, but some of them, especially those with the university education in the field, might be right. And a good system should have a mechanism to detect such persons, like a good detector must have the ability to detect rare particles, and not just classify them as noise.

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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

GeomSpinors01

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.

GeomSpinors04

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.

GeomSpinors05

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.

GeomSpinors06

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).

eomSpinors08

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.

GeomSpinors09

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.

GeomSpinors010

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.

GeomSpinors011

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.

GeomSpinors012

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.

GeomSpinors013

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.

GeomSpinors014

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.

GeomSpinors015

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

GeomSpinors016

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.

GeomSpinors017

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).

GeomSpinors018

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.

GeomSpinors019

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 ]

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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].

FiniteDescr1

FiniteDescr2

FiniteDescr3Click 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_Mfrom 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

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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

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)

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