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Matej Pavsic

Jožef Stefan Institute, Theoretical physics, Senior Researcher

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Currently, I am investigating how the geometric calculus based on Clifford algebras can be used to unify fundamental interactions and provide an insight into quantum field theories and gravity.
Address: Jožef Stefan Institute
Jamova 39
SI-1000 Ljubljana
Slovenia

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Grigoris Panotopoulos

Leonardo Gualtieri

Università degli Studi "La Sapienza" di Roma

University of Copenhagen

University of Perugia

University of Brescia

Max Planck Institute for Gravitational Physics

Aleksandr A Zheltukhin

University of Cambridge

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Papers by Matej Pavsic

On Negative Energies, Strings, Branes, and Braneworlds: A Review of Novel Approaches

On the way towards quantum gravity and the unification of interaction, several ideas have been re... more On the way towards quantum gravity and the unification of interaction, several ideas have been rejected and avenues avoided because they were perceived as physically unviable. But in the literature there are works in which it was found the contrary, namely that those rejected topics make sense after all. Such topics, reviewed in this article, are negative energies occurring in higher derivative theories and ultrahyperbolic spaces, ordering ambiguity of operators in curved spaces, the vast landscape of possible compactifications of extra dimensions in string theory, and quantization of a 3-brane in braneworld scenarios.

The Embedding Models of Induced Gravity with Bosonic Sources

Foundations of Physics 24 (1994) 1495-1518, 1994

We consider a theory in which spacetime is a 4-dimensional manifold $V_4$ embedded in an N-dimens... more We consider a theory in which spacetime is a 4-dimensional manifold $V_4$ embedded in an N-dimensional space V_N. The dynamics is given by a first order action which is a straightforward generalization of the well known Nambu-Goto string action. Instead of the latter action we then consider an equivalent action, a generalization of the Howe-Tucker action, which is a functional of the (extrinsic) embedding variables η a (x) and of the (intrinsic) induced metric g µv (x) on V4. In the quantized theory we can define an effective action by means of the Feynman path integral in which we functionally integrate over the embedding variables. What remains is functionally dependent solely on the induced metric. It is well known that the effective action so obtained contains the Ricci scalar R and its higher orders. But due to our special choice of a quantity, the so called "matter" density ω(η) in VN entering the original first order action, it turns out that the effective action contains also the source term. The latter is in general that of a p-dimensional membrane (p-brane). In particular we consider the case of bosonic point particles. Finally we discuss and clarify certain interpretational aspects of quantum mechanics from the view point of our embedding model.

A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time

International Journal of .Modern Physics A 35 (2020) 04, 2050015, 2020

An approach to the quantization of gravity in the presence matter is examined which starts from t... more An approach to the quantization of gravity in the presence matter is examined which starts from the classical Einstein-Hilbert action and matter approximated by "point" particles minimally coupled to the metric. Upon quantization, the Hamil-ton constraint assumes the form of the Schrödinger equation: it contains the usual Wheeler-DeWitt term and the term with the time derivative of the wave function. In addition, the wave function also satisfies the Klein-Gordon equation, which arises as the quantum counterpart of the constraint among particles' momenta. Comparison of the novel approach with the usual one in which matter is represented by scalar fields is performed, and shown that those approaches do not exclude, but complement each other. In final discussion it is pointed out that the classical matter could consist of superparticles or spinning particles, described by the commuting and anticommuting Grassmann coordinates, in which case spinor fields would occur after quantization. .

Pais-Uhlenbeck Oscillator and Negative Energies

International Journal of Geometric Methods in Modern Physics A, 13 (2016) no.09, 1630015, 2016

We review the occurrence of negative energies in Pais-Uhlenbeck oscillator. We point out that in ... more We review the occurrence of negative energies in Pais-Uhlenbeck oscillator. We point out that in the absence of interactions negative energies are not problematic, neither in the classical nor in the quantized theory. However, in the presence of interactions that couple positive and negative energy degrees of freedom the system is unstable, unless the potential is bounded from bellow and above. We review some approaches in the literature that attempt to avoid the problem of negative energies in the Pais-Uhlenbeck oscillator.

Manifestly Covariant Canonical Quantization of the Scalar Field and Particle Localization

Modern Physics Letters A, 2018

Particle localization within quantum field theory is revisited. Canonical quantization of a free ... more Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface Sigma, which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to Sigma. Position on Sigma is determined by a 4-vector ¯xμ. The corresponding quantum position operator, formed in terms of the operators a†(¯x), a(¯x), that create/annihilate particles on Sigma, has thus well behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with a†(¯x), the superposition coefficients forming multiparticle wave packet profiles—wave functions, including a single particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center of mass operator is introduced and its expectation values in a generic multiparticle state calculated.

A Novel View on Successive Quantizations, Leading to Increasingly More "Miraculous" States

Modern Physics Letters A, 2019

A series of successive quantizations is considered, starting with the quantization of a non relat... more A series of successive quantizations is considered, starting with the quantization of a non relativistic or relativistic point particle: 1) quantization of a particle's position, 2) quantization of wave function, 3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first order equation in time derivative for complex wave function (or functional), the equivalent second order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.

Point Particle with Extrinsic Curvature as a Boundary of a Nambu–Goto String: Classical and Quantum Model

Advances in Applied Clifford Algebras, 2015

It is shown how a string living in a higher dimensional space can be approximated as a point part... more It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe-Tucker action for such a "rigid particle" and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable e↵ective mass. The Pauli-Lubanski vector composed with such projected Dirac matrices is equal to the Pauli-Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states. arXiv:1405.7838v2 [hep-th]

On the embedding models of gravity and their prospects

Marcel Grossmann Meeting on General Relativity, 1992

Einstein's general relativity is considered as an effective theory of another, more simple theory... more

A new view about black-holes

Electron as Rigid Particle and Estimation of its Rigidity Constant

Square Root of Membrane’s Constraints

The Dirac ”square root” procedure is extended from point-particles to membranes of arbitrary dime... more The Dirac ”square root” procedure is extended from point-particles to membranes of arbitrary dimension. Membrane’s constraint which is quadratic in momenta is written as the product of two terms which are linear in momenta. This is possible if we generalize the Clifford numbers from the finite dimensional space to the infinite dimensional space. The linearized constraint is then required to annihilate a membrane’s physical quantum state. Such an equation is just a generalized Dirac equation.

arXiv version 2 of Local Tachyons

We examine the possibility of localized propagating tachyonic fields within a properly extended r... more We examine the possibility of localized propagating tachyonic fields within a properly extended relativity. A possible extension is to include superluminal transformations and reference frames. This leads to complex 4D spacetime, or real 8D spacetime M_{4,4}. The mass shell constraint in M_{4,4} becomes, after first quantization, the ultrahyperbolic Klein-Gordon equation. The Cauchy problem for such equation is not well posed, because it is not possible to freely specify initial data on a 7D hypersurface of M_{4,4}. We explicitly demonstrate that it is possible to do it on a space-like 4-surface for bradyons, and on a time-like 4-surface for tachyons. But then the evolution of a bradyonic field into the four time-like directions, or the "evolution" of a tachyonic field into the four space-like directions, is not uniquely determined. We argue that this is perhaps no so bad, because in quantum field theory (after second quantization) the classical trajectories of fields are ...

Stable Self-Interacting Pais-Uhlenbeck Oscillator

Modern Physics Letters A

It is shown that the interacting Pais-Uhlenbeck oscillator necessarily leads to a description wit... more It is shown that the interacting Pais-Uhlenbeck oscillator necessarily leads to a description with a Hamiltonian that contains positive and negative energies associated with two oscillators. Descriptions with a positive definite Hamiltonians, considered by some authors, can hold only for a free Pais-Uhlenbeck oscillator. We demonstrate that the solutions of a self-interacting Pais-Uhlenbeck oscillator are stable on islands in the parameter space, as already observed in the literature. If we slightly modify the system, by considering a sine interaction term, and/or by taking unequal masses of the two oscillators, then the system is stable on the continents that extends from zero to infinity in the parameter space. Therefore, the Pais-Uhlenbeck oscillator is quite acceptable physical system.

Geometric Spinors, Generalized Dirac Equation and Mirror Particles

It is shown that since the geometric spinors are elements of Clifford algebras, they must have th... more It is shown that since the geometric spinors are elements of Clifford algebras, they must have the same transformation properties as any other Clifford number. In general, a Clifford number $\Phi$ transforms into a new Clifford number $\Phi'$ according to $\Phi \to \Phi ' = {\rm{R}}\,\Phi \,{\rm{S}}$, i.e., by the multiplication from the left and from the right by two Clifford numbers ${\rm R}$ and ${\rm S}$. We study the case of $Cl(1,3)$, which is the Clifford algebra of the Minkowski spacetime. Depending on choice of ${\rm R}$ and ${\rm S}$, there are various possibilities, including the transformations of vectors into 3-vectors, and the transformations of the spinors of one minimal left ideal of $Cl(1,3)$ into another minimal left ideal. This, among others, has implications for understanding the observed non-conservation of parity.

Again about causality for tachyons in macrophysics

Lettere Al Nuovo Cimento Series 2, 1977

This is a second auswer to the comments by BASAN0, i.e. a brief comment on rcf. 0). Although (~ e... more

How to recover causality for tachyons even in macrophysics

Il Nuovo Cimento A, 1978

... s&amp;amp;amp;#x27;, who can, on the contrary, decide to send no signal (0i&amp;amp;a... more ... s&amp;amp;amp;#x27;, who can, on the contrary, decide to send no signal (0i&amp;amp;amp;quot; antisignal) towards s. A ]ortiori, the observer s&amp;amp;amp;#x27; can be in the impossibility of sending that signal, being lacking in information about it (for instance, s told s&amp;amp;amp;#x27;--at O--that, at time t, he, s, will send to s&amp;amp;amp;#x27; one piece of music by a ...

Clifford Algebra, Geometry and Physics

The Nature of Time: Geometry, Physics and Perception, 2003

The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. ... more The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold (C-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. Within C-space we can perform the so called polydimensional rotations which reshuffle the multivectors, e.g., a bivector into a vector, etc.. A consequence of such a polydimensional rotation is that the signature can change: it is relative to a chosen set of basis vectors. Another important consequence is that the well known unconstrained Stueckelberg theory is embedded within the constrained theory based on C-space. The essence of the Stueckelberg theory is the existence of an evolution parameter which is invariant under the Lorentz transformations. The latter parameter is interpreted as being the true time -associated with our perception of the passage of time.

On an Alternative Approach to the Relation between Bosons and Fermions: Employing Clifford Space

We further explore the idea that physics takes place in Clifford space which should be considered... more We further explore the idea that physics takes place in Clifford space which should be considered as a generalization of spacetime. Following the old observation that spinors can be represented as members of left ideals of Clifford algebra, we point out that the transformations which mix bosons and fermions could be represented by means of operators acting on Clifford algebra-valued (polyvector) fields. A generic polyvector field can be expanded either in terms of bosonic, or in terms of fermionic fields. In particular, a scalar field can transform into a mixture of bosonic and/or fermionic fields.

Higher derivative gravity and torsion from the geometry of C-spaces

Physics Letters B, 2002

We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, b... more We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension-the so called Clifford space (C-space), a manifold of points, lines, areas, etc..; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved C-space is investigated. It is shown that the curvature in C-space contains higher orders of the curvature in the underlying ordinary space. A C-space is parametrized not only by 1-vector coordinates x µ but also by the 2-vector coordinates σ µν , 3-vector coordinates σ µνρ , etc., called also holographic coordinates, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the "area" derivative ∂/∂σ µν and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates σ µν this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the C-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of M-theory.

On the wave function in a classical gravitational field

Physics Letters A, 1982

On Negative Energies, Strings, Branes, and Braneworlds: A Review of Novel Approaches

On the way towards quantum gravity and the unification of interaction, several ideas have been re... more On the way towards quantum gravity and the unification of interaction, several ideas have been rejected and avenues avoided because they were perceived as physically unviable. But in the literature there are works in which it was found the contrary, namely that those rejected topics make sense after all. Such topics, reviewed in this article, are negative energies occurring in higher derivative theories and ultrahyperbolic spaces, ordering ambiguity of operators in curved spaces, the vast landscape of possible compactifications of extra dimensions in string theory, and quantization of a 3-brane in braneworld scenarios.

The Embedding Models of Induced Gravity with Bosonic Sources

Foundations of Physics 24 (1994) 1495-1518, 1994

We consider a theory in which spacetime is a 4-dimensional manifold $V_4$ embedded in an N-dimens... more We consider a theory in which spacetime is a 4-dimensional manifold $V_4$ embedded in an N-dimensional space V_N. The dynamics is given by a first order action which is a straightforward generalization of the well known Nambu-Goto string action. Instead of the latter action we then consider an equivalent action, a generalization of the Howe-Tucker action, which is a functional of the (extrinsic) embedding variables η a (x) and of the (intrinsic) induced metric g µv (x) on V4. In the quantized theory we can define an effective action by means of the Feynman path integral in which we functionally integrate over the embedding variables. What remains is functionally dependent solely on the induced metric. It is well known that the effective action so obtained contains the Ricci scalar R and its higher orders. But due to our special choice of a quantity, the so called "matter" density ω(η) in VN entering the original first order action, it turns out that the effective action contains also the source term. The latter is in general that of a p-dimensional membrane (p-brane). In particular we consider the case of bosonic point particles. Finally we discuss and clarify certain interpretational aspects of quantum mechanics from the view point of our embedding model.

A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time

International Journal of .Modern Physics A 35 (2020) 04, 2050015, 2020

An approach to the quantization of gravity in the presence matter is examined which starts from t... more An approach to the quantization of gravity in the presence matter is examined which starts from the classical Einstein-Hilbert action and matter approximated by "point" particles minimally coupled to the metric. Upon quantization, the Hamil-ton constraint assumes the form of the Schrödinger equation: it contains the usual Wheeler-DeWitt term and the term with the time derivative of the wave function. In addition, the wave function also satisfies the Klein-Gordon equation, which arises as the quantum counterpart of the constraint among particles' momenta. Comparison of the novel approach with the usual one in which matter is represented by scalar fields is performed, and shown that those approaches do not exclude, but complement each other. In final discussion it is pointed out that the classical matter could consist of superparticles or spinning particles, described by the commuting and anticommuting Grassmann coordinates, in which case spinor fields would occur after quantization. .

Pais-Uhlenbeck Oscillator and Negative Energies

International Journal of Geometric Methods in Modern Physics A, 13 (2016) no.09, 1630015, 2016

We review the occurrence of negative energies in Pais-Uhlenbeck oscillator. We point out that in ... more We review the occurrence of negative energies in Pais-Uhlenbeck oscillator. We point out that in the absence of interactions negative energies are not problematic, neither in the classical nor in the quantized theory. However, in the presence of interactions that couple positive and negative energy degrees of freedom the system is unstable, unless the potential is bounded from bellow and above. We review some approaches in the literature that attempt to avoid the problem of negative energies in the Pais-Uhlenbeck oscillator.

Manifestly Covariant Canonical Quantization of the Scalar Field and Particle Localization

Modern Physics Letters A, 2018

Particle localization within quantum field theory is revisited. Canonical quantization of a free ... more Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface Sigma, which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to Sigma. Position on Sigma is determined by a 4-vector ¯xμ. The corresponding quantum position operator, formed in terms of the operators a†(¯x), a(¯x), that create/annihilate particles on Sigma, has thus well behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with a†(¯x), the superposition coefficients forming multiparticle wave packet profiles—wave functions, including a single particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center of mass operator is introduced and its expectation values in a generic multiparticle state calculated.

A Novel View on Successive Quantizations, Leading to Increasingly More "Miraculous" States

Modern Physics Letters A, 2019

A series of successive quantizations is considered, starting with the quantization of a non relat... more A series of successive quantizations is considered, starting with the quantization of a non relativistic or relativistic point particle: 1) quantization of a particle's position, 2) quantization of wave function, 3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first order equation in time derivative for complex wave function (or functional), the equivalent second order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.

Point Particle with Extrinsic Curvature as a Boundary of a Nambu–Goto String: Classical and Quantum Model

Advances in Applied Clifford Algebras, 2015

It is shown how a string living in a higher dimensional space can be approximated as a point part... more It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe-Tucker action for such a "rigid particle" and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable e↵ective mass. The Pauli-Lubanski vector composed with such projected Dirac matrices is equal to the Pauli-Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states. arXiv:1405.7838v2 [hep-th]

On the embedding models of gravity and their prospects

Marcel Grossmann Meeting on General Relativity, 1992

Einstein's general relativity is considered as an effective theory of another, more simple theory... more

A new view about black-holes

Electron as Rigid Particle and Estimation of its Rigidity Constant

Square Root of Membrane’s Constraints

The Dirac ”square root” procedure is extended from point-particles to membranes of arbitrary dime... more The Dirac ”square root” procedure is extended from point-particles to membranes of arbitrary dimension. Membrane’s constraint which is quadratic in momenta is written as the product of two terms which are linear in momenta. This is possible if we generalize the Clifford numbers from the finite dimensional space to the infinite dimensional space. The linearized constraint is then required to annihilate a membrane’s physical quantum state. Such an equation is just a generalized Dirac equation.

arXiv version 2 of Local Tachyons

We examine the possibility of localized propagating tachyonic fields within a properly extended r... more We examine the possibility of localized propagating tachyonic fields within a properly extended relativity. A possible extension is to include superluminal transformations and reference frames. This leads to complex 4D spacetime, or real 8D spacetime M_{4,4}. The mass shell constraint in M_{4,4} becomes, after first quantization, the ultrahyperbolic Klein-Gordon equation. The Cauchy problem for such equation is not well posed, because it is not possible to freely specify initial data on a 7D hypersurface of M_{4,4}. We explicitly demonstrate that it is possible to do it on a space-like 4-surface for bradyons, and on a time-like 4-surface for tachyons. But then the evolution of a bradyonic field into the four time-like directions, or the "evolution" of a tachyonic field into the four space-like directions, is not uniquely determined. We argue that this is perhaps no so bad, because in quantum field theory (after second quantization) the classical trajectories of fields are ...

Stable Self-Interacting Pais-Uhlenbeck Oscillator

Modern Physics Letters A

It is shown that the interacting Pais-Uhlenbeck oscillator necessarily leads to a description wit... more It is shown that the interacting Pais-Uhlenbeck oscillator necessarily leads to a description with a Hamiltonian that contains positive and negative energies associated with two oscillators. Descriptions with a positive definite Hamiltonians, considered by some authors, can hold only for a free Pais-Uhlenbeck oscillator. We demonstrate that the solutions of a self-interacting Pais-Uhlenbeck oscillator are stable on islands in the parameter space, as already observed in the literature. If we slightly modify the system, by considering a sine interaction term, and/or by taking unequal masses of the two oscillators, then the system is stable on the continents that extends from zero to infinity in the parameter space. Therefore, the Pais-Uhlenbeck oscillator is quite acceptable physical system.

Geometric Spinors, Generalized Dirac Equation and Mirror Particles

It is shown that since the geometric spinors are elements of Clifford algebras, they must have th... more It is shown that since the geometric spinors are elements of Clifford algebras, they must have the same transformation properties as any other Clifford number. In general, a Clifford number $\Phi$ transforms into a new Clifford number $\Phi'$ according to $\Phi \to \Phi ' = {\rm{R}}\,\Phi \,{\rm{S}}$, i.e., by the multiplication from the left and from the right by two Clifford numbers ${\rm R}$ and ${\rm S}$. We study the case of $Cl(1,3)$, which is the Clifford algebra of the Minkowski spacetime. Depending on choice of ${\rm R}$ and ${\rm S}$, there are various possibilities, including the transformations of vectors into 3-vectors, and the transformations of the spinors of one minimal left ideal of $Cl(1,3)$ into another minimal left ideal. This, among others, has implications for understanding the observed non-conservation of parity.

Again about causality for tachyons in macrophysics

Lettere Al Nuovo Cimento Series 2, 1977

This is a second auswer to the comments by BASAN0, i.e. a brief comment on rcf. 0). Although (~ e... more

How to recover causality for tachyons even in macrophysics

Il Nuovo Cimento A, 1978

... s&amp;amp;amp;#x27;, who can, on the contrary, decide to send no signal (0i&amp;amp;a... more ... s&amp;amp;amp;#x27;, who can, on the contrary, decide to send no signal (0i&amp;amp;amp;quot; antisignal) towards s. A ]ortiori, the observer s&amp;amp;amp;#x27; can be in the impossibility of sending that signal, being lacking in information about it (for instance, s told s&amp;amp;amp;#x27;--at O--that, at time t, he, s, will send to s&amp;amp;amp;#x27; one piece of music by a ...

Clifford Algebra, Geometry and Physics

The Nature of Time: Geometry, Physics and Perception, 2003

The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. ... more The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold (C-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. Within C-space we can perform the so called polydimensional rotations which reshuffle the multivectors, e.g., a bivector into a vector, etc.. A consequence of such a polydimensional rotation is that the signature can change: it is relative to a chosen set of basis vectors. Another important consequence is that the well known unconstrained Stueckelberg theory is embedded within the constrained theory based on C-space. The essence of the Stueckelberg theory is the existence of an evolution parameter which is invariant under the Lorentz transformations. The latter parameter is interpreted as being the true time -associated with our perception of the passage of time.

On an Alternative Approach to the Relation between Bosons and Fermions: Employing Clifford Space

We further explore the idea that physics takes place in Clifford space which should be considered... more We further explore the idea that physics takes place in Clifford space which should be considered as a generalization of spacetime. Following the old observation that spinors can be represented as members of left ideals of Clifford algebra, we point out that the transformations which mix bosons and fermions could be represented by means of operators acting on Clifford algebra-valued (polyvector) fields. A generic polyvector field can be expanded either in terms of bosonic, or in terms of fermionic fields. In particular, a scalar field can transform into a mixture of bosonic and/or fermionic fields.

Higher derivative gravity and torsion from the geometry of C-spaces

Physics Letters B, 2002

We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, b... more We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension-the so called Clifford space (C-space), a manifold of points, lines, areas, etc..; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved C-space is investigated. It is shown that the curvature in C-space contains higher orders of the curvature in the underlying ordinary space. A C-space is parametrized not only by 1-vector coordinates x µ but also by the 2-vector coordinates σ µν , 3-vector coordinates σ µνρ , etc., called also holographic coordinates, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the "area" derivative ∂/∂σ µν and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates σ µν this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the C-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of M-theory.

On the wave function in a classical gravitational field

Physics Letters A, 1982

Stumbling Blocks Against Unification

World Scientific, 2020

The history is full of misconceptions that opposed the progress of physics. The book starts with ... more The history is full of misconceptions that opposed the progress of physics. The book starts with reviewing some historical cases, such as the arguments against the Earth rotation, or the famous problem of ¾ in the theory of electromagnetic mass of electron. After having pointed out that misconceptions have been common in the history of physics, it is argued that they must be present today as well. In fact, it is now commonly being realized that in the last forty years there has been no significant progress in the fundamental theoretical physics. A reason certainly lies in certain stumbling blocks on our way towards the unification of interaction and of gravity with quantum mechanics. The author discusses what he perceives as some persisting misconceptions that have not yet been recognized as such by physics community in general.