6 research outputs found
Positive Cosmological Constant and Quantum Theory
We argue that quantum theory should proceed not from a spacetime background
but from a Lie algebra, which is treated as a symmetry algebra. Then the fact
that the cosmological constant is positive means not that the spacetime
background is curved but that the de Sitter (dS) algebra as the symmetry
algebra is more relevant than the Poincare or anti de Sitter ones. The physical
interpretation of irreducible representations (IRs) of the dS algebra is
considerably different from that for the other two algebras. One IR of the dS
algebra splits into independent IRs for a particle and its antiparticle only
when Poincare approximation works with a high accuracy. Only in this case
additive quantum numbers such as electric, baryon and lepton charges are
conserved, while at early stages of the Universe they could not be conserved.
Another property of IRs of the dS algebra is that only fermions can be
elementary and there can be no neutral elementary particles. The cosmological
repulsion is a simple kinematical consequence of dS symmetry on quantum level
when quasiclassical approximation is valid. Therefore the cosmological constant
problem does not exist and there is no need to involve dark energy or other
fields for explaining this phenomenon (in agreement with a similar conclusion
by Bianchi and Rovelli).Comment: 44 pages, Latex, no figures. A revised version published in Symmetry,
Special Issue: Quantum Symmetr
A New Look at the Position Operator in Quantum Theory
The postulate that coordinate and momentum representations are related to
each other by the Fourier transform has been accepted from the beginning of
quantum theory by analogy with classical electrodynamics. As a consequence, an
inevitable effect in standard theory is the wave packet spreading (WPS) of the
photon coordinate wave function in directions perpendicular to the photon
momentum. This leads to several paradoxes. The most striking of them is that
coordinate wave functions of photons emitted by stars have cosmic sizes and
strong arguments indicate that this contradicts observational data. We argue
that the above postulate is based neither on strong theoretical arguments nor
on experimental data and propose a new consistent definition of the position
operator. Then WPS in directions perpendicular to the particle momentum is
absent and the paradoxes are resolved. Different components of the new position
operator do not commute with each other and, as a consequence, there is no wave
function in coordinate representation. Implications of the results for
entanglement, quantum locality and the problem of time in quantum theory are
discussed.Comment: 68 pages. A version published in Physics of Particles and Nuclei has
been considerably revised in view of our discussions with Anatoly Kamchatnov
and his constructive criticis
Symmetries in Foundation of Quantum Theory and Mathematics
In standard quantum theory, symmetry is defined in the spirit of Klein's
Erlangen Program: the background space has a symmetry group, and the basic
operators should commute according to the Lie algebra of that group. We argue
that the definition should be the opposite: background space has a direct
physical meaning only on classical level while on quantum level symmetry should
be defined by a Lie algebra of basic operators. Then the fact that de Sitter
symmetry is more general than Poincare one can be proved mathematically. The
problem of explaining cosmological acceleration is very difficult but, as
follows from our results, there exists a scenario that the phenomenon of
cosmological acceleration can be explained proceeding from basic principles of
quantum theory. The explanation has nothing to do with existence or
nonexistence of dark energy and therefore the cosmological constant problem and
the dark energy problem do not arise. We consider finite quantum theory (FQT)
where states are elements of a space over a finite ring or field with
characteristic and operators of physical quantities act in this space. We
prove that, with the same approach to symmetry, FQT and finite mathematics are
more general than standard quantum theory and classical mathematics,
respectively: the latter theories are special degenerated cases of the former
ones in the formal limit .Comment: 32 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1905.02788, arXiv:1104.464
Finite Mathematics, Finite Quantum Theory and Applications to Gravity and Particle Theory
We argue that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete and even finite, is described by classical mathematics involving the notions of infinitely small, continuity etc. Moreover, since classical mathematics has its own foundational problems which cannot be resolved (as follows, in particular , from Gödel's incompleteness theorems), the ultimate physical theory cannot be based on that mathematics. In the first part of the work we discuss inconsistencies in standard quantum theory and reformulate the theory such that it can be naturally generalized to a formulation based on finite mathematics. It is shown that: a) as a consequence of inconsistent definition of standard position operator, predictions of the theory contradict the data on observations of stars; b) the cosmological acceleration and gravity can be treated simply as kinematical manifestations of quantum de Sitter symmetry, i.e. the cosmological constant problem does not exist, and for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed. In the second part we first prove that classical mathematics is a special degenerate case of finite mathematics in the formal limit when the characteristic p of the field or ring in the latter goes to infinity. This result fundamentally changes the standard paradigm on what mathematics and what physics are the most fundamental. Then we consider a quantum theory based on finite mathematics with a large p. In this approach the de Sitter gravitational constant depends on p and disappears in the formal limit p → ∞, i.e. gravity is a consequence of finiteness of nature. The application to particle theory gives that the notion of a particle and its antiparticle is only approximate and, as a consequence: a) the electric charge and the baryon and lepton quantum numbers can be only approximately conserved; b) particles which in standard theory are treated as neutral (i.e. coinciding with their antiparticles) cannot be elementary. We argue that only Dirac singletons can be true elementary particles and the main reasons are: a) massless and massive particles can be constructed from singletons; b) while irreducible representations describing massless and massive particles are necessarily over a field, irreducible representations describing singletons can be constructed over a ring. Finally we discuss a conjecture that classical time t manifests itself as a consequence of the fact that p changes, i.e. p and not t is the true evolution parameter