Description of Acivity

Matej Pavsic J. Stefan Institute, Ljubljana, Slovenia

In 70’s I was fascinated by all sorts of transformations, Lorentz transformations, space inversion, time reversal, scale and conformal transformations.

Mirror particles

In the paper External Inversion, Internal Inversion, and Reflection Invariance Int. J. Theor. Phys. 9, 229 (1974) I discussed a theory based on the postulate that Nature should be invariant under space inversion; a consequence of such a postulate is the prediction of mirror particles which imply that there should exist a mirror matter. Such exact parity model and its consequences, especially in astrophysics, are nowadays extensively studied by many authors. Such idea was first proposed by T. D. Lee and C.N. Yang in his 1956 paper on parity non conservation Phys. Rev. 104, 254 (1956) and was further elaborated in 1966 by Yu. Kobzarev, L.B. Okun and I.Ya. Pomeranchuk Soviet J. Nucl. Phys. 5, 837 (1966).

Scale transformations

The scale transformations contain as a particular case (when the scale factor is -1) the superluminal transformations. In the local journal Obzornik za Matematiko in Fiziko 19, 20 (1973), I published a paper in which I considered improper Lorentz transformation and its extension to superluminal transformations. Before that, in January 1971, I also wrote a longer and more complete paper The Extended Special Theory of Relativity  which I send to Philosophical Magazine, but -being inexperienced (I was still an undergraduate student)- I did not include an accompanying letter requesting publication, so I have never received an answer. In the subsequent years, a similar theory was independently published in a series of papers by E. Recami and R. Mignani, culminating in the review articles Rivista del Nuovo Cimento 4, 209 (1974);  9, 1 (1986). At that time I was already well aware that superluminal transformations, if taken actively, imply the existence of particles-tachyons-that travel faster than light, and that such particles would lead to the well known causality paradoxes. In the paper Towards Understanding Quantum Mechanics, General Relativity and the Tachyonic causality paradoxes Lett. N. Cim. 30, 111 (1981) I discussed my view of how to resolve the problem. For that aim I had to adopt the Everett many worlds interpretation of quantum mechanics. The resolution of the tachyonic causality paradoxes is analogous to the solution proposed in 1991 by D. Deutsch Physical Review D 44, 3197 (1991) for worm holes which behave as time machines. Besides causality, there are other problems with tachyons, occurring mainly in field theory, which have convinced most researchers that tachyons do not exist. But, as it has been realized by a number of researchers , such problems do not exist in the Fock-Stueckelberg theory of the relativistic particle and its quantization. The latter theory was initially also investigated by Feynman, but later, because of the opposition of many respectable physicists, he abandoned it (and only included its exposition in the appendix of his paper Physical Review 80, 440 (1950)). See a nice historic exposition by S.S. Schweber: Feynman and the visualization of space-time process Reviews of Modern Physics 58 449-505 (1986).

In 1976/77 I spent one year at the Istituto di Fisica Teorica, Università di Catania, Italy, where I have written a number of papers in collaboration with E.Recami, P.Caldirola and V. de Sabatta. There I also prepared my PhD thesis Unified Theory of Gravitation and Electromagnetism, based on the Conformal Group SO(4,2) Nuovo Cimento B 41, 397-427 (1977) in which I considered the active (generalized) conformal transformations. They imply that size of an object is not fixed, but is a dynamical degree of freedom. The idea is similar to the one by Weyl who considered the geometry in which the sizes of objects depends on path. But there is a big difference, since in my approach the size of a free object may change even in flat spacetime (like position of a moving object may change). However, the size (scale) of a bounded microscopic object obeying the laws of quantum mechanic cannot have arbitrary continuous expectation value (analogously, the position coordinates of an electron around a nucleus cannot have arbitrary continuous expectation values). In the paper An Attempt to Resolve the Astrophysical Puzzles by Postulating Scale Degree of Freedom Int. J. Theor. Phys. 14, 299 (1975) I considered the implications of active scale and conformal transformations in astrophysics. I elaborated the idea of the active scale transformations further in a number of papers including Introducing the Dilatational Degree of Freedom: Special Relativity  in V  J. Phys. A 13, 1367-1387 (1980).

Spacetime as a membrane in higher dimensions

In 80′ I started to work on the idea that our spacetime is a 4-dimensional surface swept by a 3-brane moving in a higher dimensional embedding space. I first explained the idea in its rough contours in the paper Towards Understanding Quantum Mechanics, General Relativity and the Tachyonic causality paradoxes Lett. N. Cim. 30, 111 (1981) mentioned before. A much more elaborated theory was published in the papers On the Quantization of Gravity by Embedding Spacetime in a Higher Dimensional Space Class. Quant. Grav. 2, 869-889 (1985). and Classical Theory of a Spacetime Sheet Phys. Lett. A 107, 66-70 (1985). Those papers were more or less still along the lines of the Regge-Teitelboim approach in which the Einstein-Hilbert action was written in terms of the extrinsic, embedding, coordinates. But a hint of a new approach -which is similar to the modern brane world scenario– is already touched in that papers. Next year I published two papers Einstein’s Gravity from a First Order Lagrangian in an Embedding Space Phys. Lett. A 116, 1-5 (1986) and String Model for General Relativity and Canonical Formalism for Minimal Surfaces Nuovo Cimento A 95, 297-310 (1986) in which the 3-brane action was just that of a minimal surface, i.e., the Dirac-Nambu-Goto action, and I abandoned considering the Regge-Teitelboim action at all. So I proposed that our world was just a 3-brane moving according to the minimal surface action principle and sweeping a four dimensional surface which was identified with our observed spacetime.  Moreover, I proposed that matter on our 3-brane was a result of the intersection of our brane with other branes. I showed that in a particular case the intersection is just a point particle whose worldline -as it follows from the action principle- is a geodesic in the 4-surface swept by our 3-brane.

Recently, in my book The Landscape of Theoretical Physics: A Global View  (Kluwer, 2001) and in the paper A Brane World Model with Intersecting Branes  Phys. Lett. A 283, 8-14 (2001) I generalized this result to the case when the intersection is a p-brane of any p and found that the (p+1)-dimensional world sheet swept by the p-brane is a minimal surface. The later situation, of course holds when our world is not a 3-brane but a higher dimensional brane. The extra dimensions of our brane world could then be responsible -via Kaluza-Klein mechanism- for other interactions, like the electromagnetic one, etc. Moreover, I have shown that matter in our brane world can be due to the brane’s self intersections as well. The self intersection can be topologically highly non trivial so that, e.g.,  a bound system may consist of three particles which cannot be asymptotically free, although at short distances they behave as free particles.


In the period 1987-1994 I published a series of papers in which I was exploring p-branes within the framework of the conventional theory. In that period I also collaborated with prof. A.O. Barut (University of Colorado, Boulder, USA) on the problem of classical and quantum spinning particles, and electrically charged membranes.

Fock-Stueckelberg-Feynman-Schwinger approach with an invariant evolution parameter

All the time I was aware that quantization of p-branes was a rather tricky job. In order to circumvent the difficulties I started to study the Fock-Stueckelberg-Feynman-Schwinger approach to description of relativistic particles and extended it to p-branes. In such a theory there are no constraints on the coordinates and momenta so that quantization is straightforward. The theory, of course, deviates from the conventional theory of p-branes, but I showed that it contains the conventional p-branes as a particular case. My papers on that subject are here.

Clifford algebras and Clifford space

In 1992 I met in Catania prof. W.A. Rodrigues, jr., who introduced me into the subject of Clifford algebra. We were both guest of E. Recami. Our aim was to collaborate in a joint project on various models of the spinning particle and find the connection between the Barut-Zanghi model and its reformulation by means of Clifford algebra. So prof. Rodrigues started to talk me about Clifford algebra as a useful tool for geometry and physics. After two weeks of discussion I became a real enthusiast and since then I kept on exploring how physical theories could be formulated in the framework of geometric calculus based on Clifford algebra. So I have found that if we generalize point particle and p-brane theory to polyvectors (which are elements of Clifford algebra, e.g., scalars, 1-vectors, 2-vectors, etc.), then the additional degree of freedom, namely the (pseudo) scalar degree of freedom, renders the object’s spacetime coordinates and momenta unconstrained, just as in the Stueckelberg theory! The Stueckelberg, unconstrained, theory of point particles and p-branes is embedded in the straightforward, natural, generalization of the usual, constrained, theory.  The generalized theory is just like the usual, constrained, relativity, but it acts in Clifford space. Those results can be found in my book The Landscape of Theoretical Physics: A Global View; From Point Particles to the Brane World and Beyond, in Search of a Unifying Theory (Kluwer, 2001) and in the paper Clifford Algebra Based Polydimensional Relativity and Relativistic Dynamics which I presented at the IARD 2000 Conference, 26-28 June, 2000, Tel Aviv (Foundation of Physics 31, 1185-1209 (2001)).

In the paper Clifford Space as the Arena for Physics  presented at the  IARD 2002 Conference, 24-26 June, 2002, Washington, DC (Foundations of Physics 33, 1277-1306 (2003)) I proposed and developed the idea that extended objects can be modeled  in Clifford space.  The points of Clifford space are described by the coordinates which from the view point of flat Minkowski space  M  are components of a scalar, vector, bivector, pseudovector and pseudoscalar, and I assigned to those coordinates the physical meaning of the generalized center of mass coordinates. In particular, the extended objects can be just the  closed branes (p-branes).  For instance, in the case of closed strings we have a 2-dimensional surface enclosed by a 1-dimensional line. Integrating over the oriented area, we obtain a finite effective oriented area given in terms of bivector coordinates x^{\mu \nu}.Such bivector coordinates provide an approximate description of a closed string; they do not provide a complete description of the string, but nevertheless they provide a better approximation than the mere center of mass coordinates. Analogous holds for higher grade objects. For a more precise description see

Though many physicists are interested in the geometric calculus based on Clifford algebra (as developed by D. Hestenes), its full power for theoretical physics is not yet generally appreciated. Those who are familiar with geometric calculus somehow fail to fully exploit it and naturally generalize the existing physical theories from spacetime to a larger manifold, so called Clifford space or shortly C-space. The  “points” of flat C-space can be described by Clifford algebra valued coordinates. Since (as it is well known)  spinors can be represented as the elements of left or right ideals of Clifford algebra, in C-space spinors occur automatically.  This opens a promising possibility for a natural reformulation of string theory. In  string theory the variables that describe a string are not only the so called “bosonic” coordinates (i.e., the usual spacetime coordinates), but also the so called “fermionic” coordinates (usually taken to be Grassmannian). I believe that such a C-space approach to string and brane theory will enable us to (re)formulate the conjectured M-theory which is a hot topics in string theory research.

In 2005 I published a paper Clifford Space a Generalization of Spacetime: Prospects for QFT of point Particles and Strings Foundations of Physics 35 , 1617-1642 (2005) in which I generalized string theory to C-space.  So it turns out that we do not need a higher dimensional target spacetime for a consistent formulation of string theory. Instead of a higher dimensional space we have 16-dimensional Clifford space which also provides a natural framework for description of superstrings and supersymmetry,  since-as said above-spinors are just the elements of left (or right) minimal ideals of Clifford algebra. Although Clifford space is itself a higher dimensional space, the status of its extra dimensions is completely different from the status of extra spacetime dimensions. The extra dimensions of Clifford space are related to extension of objects, their effective oriented areas, volumes, etc., as indicated in previous paragraph. So they are in a sense analogous to the dimensions of 3N-dimensional configuration space of, e.g.,  a system of N-point particles; all those N particles live in 4-dimensional spacetime.  Because of the tension, the ordinary Dirac-Nambu-Goto p-branes  cannot freely expand or shrink their p-volumes. Therefore their  description in C-space requires introduction of curved  C-space. In flat C-space one can describe tensionless branes.

In the paper Higher Derivative Gravity and Torsion from the Geometry of C-Spaces Physics Letters B 539, 133-142 (2002)  (with Carlos Castro)  we explore in detail some interesting properties of  curved C-space. In another paper Clifford Algebra of Spacetime and the Conformal Group   (with C. Castro) International Journal of Theoretical Physics 42, 1693-1705 (2003)  we have  found that  16-dimensional C-space  (belonging to 4-dimensional spacetime) contains  a 6-dimensional subspace with the signature (+ - - - +)  whose isometry  group is SO(4,2) . The latter group is isomorphic to the conformal group of spacetime.

Another very nice application of Clifford algebra is explored in the paper How the Geometric Calculus Resolves the Ordering Ambiguity of Quantum Theory in Curved space  Classical and Quantum Gravity 20, 2697-2714 (2003). Usually momentum in quantum mechanics is defined as the partial derivative with respect to coordinates. This works well in flat space, but in curved space there are several well known difficulties, the most notorious being the ordering problem. Namely, there is an ambiguity of how to construct the Hamiltonian which is the quadratic form of momenta that includes position dependent metric tensor. In geometric calculus the momentum operator is defined as the vector derivative (the gradient); it can be expanded in terms of basis vectors (which are position dependent). The product of two such operators is unambiguous, and such is the Hamiltonian which is just the D’Alambert operator in curved space; the curvature term is not present in the Hamiltonian if we restrict our consideration to the ordinary space (bosonic coordinates only), and do not consider the full C-space (which, as we indicated above, contains fermionic coordinates).

In 2002 I started to investigate a theory in which a 16-dimensional curved Clifford space provides a realization of Kaluza-Klein theory. No extra dimensions of spacetime are needed:  “extra dimensions” are in C-space.  We explore the spin gauge theory in C-space and show that the generalized spin connection contains the usual 4-dimensional gravity and Yang-Mills fields of the U(1)xSU(2)xSU(3) gauge group. The representation space for the latter group is provided by 16-component generalized spinors composed of four complex 4-component spinors, defined geometrically as the members of four independent left minimal ideals of Clifford algebra. These ideas appeared in the paper Kaluza-Klein Theory without Extra Dimensions: Curved Clifford Space Physics Letters B 614, 85-95 (2005) (see also Clifford Space as a Generalization of Spacetime: Prospects for Unification in Physics hep-th/0411053  and Spin Gauge Theory of Gravity in Clifford Space: A Realization of Kaluza-Klein Theory in 4-dimensional Spacetime   Int. J.  Mod. Phys. A  21 (2006) 5905-5956).

A spin-off of the above research is a realization that the extra degrees of freedom (extra “structure”) entering the generalized spinorial fields contains a subset of degrees of freedom that change under space inversion. The latter degrees of freedom can thus distinguish particles from mirror particles. In my 1974 paper External Inversion, Internal Inversion, and Reflection Invariance Int. J . Theor. Phys. 9, 229 (1974)  I did not specify what the extra degrees of freedom are; I denoted them by a single symbol alpha, and considered a field psi(x^mu,alpha) that depend not only on spacetime co-ordinates \(x^mu\), but also on the  additional parameter alpha. In paper Space inversion of spinor revisited: A possible explanation of chiral behavior in weak interactions   Physics Letters B 692, 212-217 (2010) I propose a model in which under parity a spinor of one left ideal transforms into a spinor of another left ideal. This brings a novel insight into the role of chirality in weak interactions.

In  the paper A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras  Advances in Applied Clifford Algebras DOI:10.1007/s00006-011-0314-4 I discuss the spaces whose elements are orthogonal and symplectic vectors. The ordinary phase space, whose elements are symplectic vectors, can be generalized to a super phase space that contains orthogonal vectors as well. Symplectic basis vectors satisfy the Heisenberg commutation relations, whereas orthogonal basis vectors, if transformed into the Witt basis, satisfy the fermionic commutation relations. The commuting coordinates and the symplectic basis vectors span the bosonic subspace of the phase space, whereas the anticommuting (Grassmann) coordinates and the orthogonal basis vectors span the fermionic subspace. The Poisson brackets between commuting coordinates of phase space turn out to be equal to the commutators between the symplectic basis vectors; the Poisson brackets between the anticommuting coordinates of phase space are equal to the anticommutators between the orthogonal basis vectors. Usually, by `quantization’ it is understood the replacement of classical coordinates of phase space by corresponding operators, the Poisson bracket relations being replaced by the corresponding commutation or anticommutation relations. The insight gained in this approach (that, to my knowledgement, has not been explicitly articulated so far)  is that quantum mechanical operators are in fact the symplectic or orthogonal basis vectors that are present already in the classical description of phase space. If we rewrite the classical equations of motion for coordinates and momenta into an equivalent form that involves the time derivative of the basis vectors, and then assume that coordinates and momenta can trace any trajectory in phase space, then it turns out that the basis vectors satisfy the Heisenberg equations of motion. I have thus pointed out how `quantization’ can be seen from yet another perspective. Moreover, I have reformulated and generalized the theory of quantized fields. Finally, I showed how the fact that the basis vectors on the one hand are quantum mechanical operators, and on the other hand they give metric, could be exploited in the development of quantum gravity.

If we consider a 6-dimensional space with signature (2,4) (i.e., two time-like and four space-like dimensions), then, by employing the light-cone coordinates, we obtain the Stueckelberg theory. This is shown in the paper On the Stueckelberg Like Generalization of General Relativity   J. Phys. Conf. Ser. 330 (2011) 01201 .    What about ghosts? It is usually taken for granted that time like dimensions imply ghosts. But there is another, not so well known, possibility that is based on an alternative definition of vacuum  [Cangemi D, Jackiw R and Zwiebach B 1996 Annals of Physics 245 408; Benedict E, Jackiw R and Lee H J 1996 Phys. Rev. D 54 6213], in which case no ghosts are associated with time like dimensions. How this works within the context of string theory and quantum field theory, and how this can resolve the cosmological constant problem, was shown in  Found. Phys. 35 1617 (2005)  (Preprint hep-th/0501222);  Phys. Lett. A 254 119 (1999) (Preprint hep-th/9812123); The Landscape of Theoretical Physics: A Global View; From Point Particles to the Brane World and Beyond, in Search of a Unifying Theory (Kluwer, 2001). For the 6-dimensional space we can take a subspace of Clifford space.

The Clifford space of the 4D Minkowski spacetime has  signature (8,8). The spaces with more than one time-like dimension are called ultrahyperbolic spaces. Such spaces are usually considered as unsuitable for physics for two main reasons. (i) They imply the occurrence of negative energies in the classical theory. In the quantized theory, depending on the choice of vacuum we have either the states with negative energies and positive probabilities, or the states with positive energies and negative probabilities (ghosts). (ii) The Cauchy problem cannot be well posed. The latter case was addressed in the paper Localized Propagating Tachyons in Extended Relativity Theories  Adv. Appl. Clifford Algebras 23 (2013) 469-495  DOI: 10.1007/s00006-013-0381-9  e-Print: arXiv:1201.5755 [hep-th] .
The former case was considered in PseudoEuclidean signature harmonic oscillator, quantum field theory and vanishing cosmological constant  Phys. Lett. A254 (1999) 119-125  DOI: 10.1016/S0375-9601(99)00145-0  e-Print: hep-th/9812123 . In the absence of interactions, negative energy states present no problem (see also R.P. Woodard, Lect. Notes Phys. 720 (2007) 403). But in the presence of interactions between positive and negative energy states, the system can become unstable. Common belief is that such interacting systems are necessarily unstable. But it has been shown that this is not necessarily so [Quantum Field Theories in Spaces with Neutral Signatures J.Phys.Conf.Ser. 437 (2013) 012006 DOI: 10.1088/1742-6596/437/1/012006 e-Print: arXiv:1210.6820 [hep-th] ].

Further possible physical implications of Clifford space are:
(i) Explanation of quasicrystals
Quasicrystals, with crystallographically forbidden symmetries, cannot be explained in terms of local interactions in three dimensions. They can be explained as regular crystals in 6-dimensions, projected in 3-dimensions. The Clifford space C, associated with objects in spacetime, has sixteen dimensions, and one can envisage that there exist in C the crystals that, from the 3D point of view, appear as quasicrystals (see J.Phys.Conf.Ser. 437 (2013) 012006 DOI: 10.1088/1742-6596/437/1/012006 e-Print: arXiv:1210.6820 [hep-th]).

   (ii) Emergence of Big Bang
In a field theory with neutral signature, there is an outburst of positive and negative energy particles, like an explosion, that is eventually stabilized (see
Quantum Field Theories in Spaces with Neutral Signatures and Pais-Uhlenbeck oscillator and negative energies. This is reminiscent of the Big Bang. Our universe indeed seems to emerge in an explosion. But in our universe we do not “see” equal number of positive and negative energy particles. Can then such a quasi unstable vacuum be an explanation for Big Bang?

Description of our universe requires fermions and accompanying gauge fields, including gravitation. According to the Clifford algebra generalized Dirac equation (Dirac-Kähler equation) there are four sorts of the 4-component spinors, with energy signs a shown in eq. (111) of paper J.Phys.Conf.Ser. 437 (2013) 012006 DOI: 10.1088/1742-6596/437/1/012006 e-Print: arXiv:1210.6820 [hep-th]. The vacuum of such field has vanishing energy and evolves into a superposition of positive and negative energy fermions, so that the total energy is conserved. A possible scenario is that the branch of the superposition in which we find ourselves, has the sea of negative energy states of the first and the second, and the sea positive energy states of the third and forth minimal left ideal of Cl(1, 3). According to the papers Space inversion of spinor revisited: A possible explanation of chiral behavior in weak interactions   Physics Letters B 692, 212-217 (2010) and Geometric Spinors, Generalized Dirac Equation and Mirror Particles , presented at conference  C13-06-24.8 , the former states are associated with the familiar, weakly interacting particles, whereas the latter states are associated with mirror particles, coupled to mirror gauge fields, and thus invisible to us. According to the field theory based on the Dirac-Kähler equation, the unstable vacuum could be an explanation for Big Bang.

  (iii) Emergence of excess heat
In the case of the classical pseudoeuclidean oscillator, we have found that certain interactions prevent the runaway solutions, and make the system stable. Collisions of an otherwise unstable pseudoeuclidean oscillator with surrounding particles also stabilize the oscillator. After such collisions, the surronding particles gain kinetic energy. A material made of such oscillators would thus increase the temperature of the surrounding medium, after being immersed into it. This is very hypothetical, but the history of physics teaches us that we can never be sure about what surprises are waiting ahead of us. Schechtman’s discovery of quasicrystals  was ridiculed, because, in view of the established crystalographic theory, it was considered as impossible. Fortunately, that was a simple experiment and it was not difficult for other labs to repeat it.


Higher derivative theories and negative energies

Negative energies also occur in higher derivative theories. Higher derivative theories are very important for quantum gravity, but, because of the presence of negative energies, they are generally considered as very problematic, if not completely unsuitable for physics. Negative energies arise from the wrong signs of certain terms in the Ostrogradsky Hamiltonian. In a quantized theory, such wrong signs can manifest themselves in the presence of ghost states that break unitarity. With an alternative quantization procedure, based on a different choice of vacuum, cited above, one has negative energy states, just as in the classical higher derivative theory, and no ghost states. A toy model for higher-derivative theories is the Pais-Uhlenbeck (PU) oscillator.  It has been a common believe that, because of the presence of negative energy states, the self-interacting PU oscillators is unstable. But the authors S. M. Carroll and M. Hoffman, Phys. Rev. D 68, 023509 (2003), A. V. Smilga, Nucl. Phys. B 706, 598 (2005) [hep-th/0407231], A. V. Smilga, Phys. Lett. B 632, 433 (2006) [hep-th/0503213], I. B. Ilhan and A. Kovner, arXiv:1301.4879 [hep-th], and A. V. Smilga, SIGMA 5, 017 (2009) [arXiv:0808.0139 [quant-ph]],  have found that for small initial velocities and coupling constants there exist islands of stability. Moreover, an example of an unconditionally stable interacting system was found by A. V. Smilga, SIGMA 5, 017 (2009) [arXiv:0808.0139 [quant-ph]]. This system  is a non linear extension of the PU oscillator. Further, if to the ordinary, linear, PU oscillator we add a self-interaction term that is bounded from below and from above, such as \frac{\lambda}{4} sin^4 x, then, as shown in  Stable Self-Interacting Pais-Uhlenbeck Oscillator  Modern Physics Letters A 28, 1350165 (2013) DOI:10.1142/S0217732313501654 [ arXiv:1302.5257 [gr-qc]] such a system is stable for any value of initial velocity, and is thus an example of a viable higher derivative theory.

In the paper  Pais-Uhlenbeck Oscillator with a Benign Friction Force  Physical Review D 87, 107502 (2013)  DOI:10.1103/PhysRevD.87.107502 I show that the PU oscillator can be stable even in the presence of damping. This is so, because damping does not arise from the first order term only, but also from the third order one. In a special case, in which the two damping constants have opposite signs,  we have the unstable oscillator considered by V.V. Nesterenko, Physical Review D 75, 087703 (2007).DOI:10.1103/PhysRevD.75.087703.

Klein-Gordon-Wheeler-DeWitt Schroedinger Equation

In the  paper Klein-Gordon-Wheeler-DeWitt Schroedinger Equation Phys. Lett. B703 (2011) 614-619 DOI:10.1016/j.physletb.2011.08.041
we consider the system that consists of an object coupled to the gravitational field. We describe the object by its center of mass coordinatesX^\mu, and neglect the object’s structure and extension. The action  for such a system is the gravitational (Einstein-Hilbert) action plus  the point particle action. Such description is valid outside the extended obejct, but it breaks down at smaller distances. Because the particle is not a black hole, its radius is greater than the Schwarzschild radius. We thus consider a “point particle” coupled to the gravitational field. The classical constraints become after quantizations a system of equations that comprises the Klein-Gordon, Wheeler-DeWitt and Schroedinger equation. The notorious “problem of time” does not occur in this approach, because the particle’s coordinate X^0 is time. The wave function(al) \Psi[X^\mu,q_{ij}], satisfying the Klein-Gordon equation, is a generalization of the Klein-Gordon field that depends on X^\mu only. In quantum field theory, the usual Klein-Gordon field, \Psi(X^\mu), after the second quantization, becomes an operator field that, roughly speaking, creates and annihilates particles at spacetime points X^\mu. Analogously, we can envisage, that the function(al) \Psi[X^\mu,q_{ij}] should also be considered as a field that can be (secondly) quantized and promoted to an operator that creates or annihilates a particle (in general, a p-brane) at X^\mu, together with the gravitational field q_{ij}. We have thus a vision that the quantum field theory of a scalar or spinor field in the presence of a gravitational field could be formulated differently from what we have been accustomed so far. In this paper, we investigated an  approach, in which the classical action was I [X^\mu (\tau),N,N^i,q_{ij} (x)],   and, after quantizing it, we arrived at the wave functional \Psi[X^\mu,q_{ij}], i.e., a generalized field that did not depend only on the particle’s position X^\mu in spacetime, but also on the dynamical variables of gravity, q_{ij} (x). Quantum field theory of the generalized field \Psi[X^\mu,q_{ij}] is an alternative to the usual quantum field theoretic approaches to gravity coupled to matter. Since the usual approaches have not yet led us to a consistent theory of quantum gravity, it is worth to investigate what will bring the new approach, conceived in this paper.

One possible extension of  the above approach is to consider 5-dimensional spacetime. Then, besides gravity we obtain the electromagnetism via the Kaluza-Klein mechanism. This is done in Wheeler-DeWitt Equation in Five Dimensions and Modified QED Phys.Lett. B717 (2012) 441-446 DOI: 10.1016/j.physletb.2012.09.034  arXiv:1207.4594  [gr-qc].

Another possible extension is in taking the 6-dimensional space (e.g., a subspace of Clifford space) and consider the light-cone coordinates for the 5th and 6th dimensions. Then we obtain the Stueckelberg theory in the presence of gravity. I investigted this appraoch in the paper  On the Stueckelberg Like Generalization of General Relativity   J.Phys.Conf.Ser. 330 (2011) 012011 DOI: 10.1088/1742-6596/330/1/012011   arXiv:1104.2462 [math-ph] .

Matej Pavsic
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