Lorentz invariance in loop quantum gravity

Last updated June 18, 2024

In relativistic physics, Lorentz invariance states that the laws of physics should remain unchanged under Lorentz transformation. In quantum gravity, Lorentz invariance measures the universal features in the hypothetical loop quantum gravity universes; which is a hypothetical theory that explains the quantum theory of gravity based on a geometrical interpretation of the theory of relativity. The various hypothetical design models for the universe, multiverse, and loop quantum gravity could have various general covariant principle results.

Grand Unification Epoch

The Grand Unification Epoch is the era in time in the chronology of the universe where no elementary particles existed; the three gauge interactions of the Standard Model, which define the electromagnetic and weak and/or strong interactions or forces, are merged into one singular force. Scientific consensus suggests that 3 minutes after the Big Bang: protons and neutrons began to come together to form the nuclei of simple elements.^[1] Loop quantum gravity theories, in contrast, place the origin and subsequently, the age of elementary particles, and the age of Lorentz invariance, beyond 13.799 ± 0.021 billion years ago.

The permanence of Lorentz invariance constants is based on elementary particles and their features. There are eons of time before the Big Bang to build the universe from black holes and older multiverses. There is a selective process that creates features in elementary particles, such as accepting, storing, and giving energy. Lee Smolin's books about loop quantum gravity posit that this theory contains the evolutionary ideas of "reproduction" and "mutation" of universes, elementary particles, as well as being formally analogous to models of population biology.^{[ citation needed ]}

Earlier universes

In the early universes before the Big Bang, there are theories that ''loop quantum gravity" and "loop quantum structures'' formed space. The Lorentz invariance and universal constants describe elementary particles that do not yet exist.

A fecund universe is a multiverse theory by Lee Smolin about the role of black holes. The theory suggests that black holes and loop quantum gravity connected early universes together; that loop quantum gravity can be pulled into black holes, and that within Fecund universes each new universe has slightly different laws of physics. Because these laws are only slightly different, each is assumed to be like a mutation of the early universes.

Minkowski spacetime

Loop quantum gravity (LQG) is a quantization of a classical Lagrangian classical field theory. It is equivalent to the Einstein–Cartan theory in that it leads to the same equations of motion describing general relativity with torsion.

Global Lorentz invariance is broken in LQG as it is broken in general relativity (with the exception of Minkowski spacetime, which is one particular solution of the Einstein field equations). Alternatively, there has been much talk about possible local and global violations of Lorentz invariance beyond those expected in straightforward general relativity.

Further research into whether the LQG analogue of Minkowski spacetime breaks or preserves global Lorentz invariance is needed and Carlo Rovelli and coworkers have recently been investigating the Minkowski state of LQG using spin foam techniques. It is expected that these questions will all remain open as long as the classical limits of various LQG models (see below for the sources of variation) cannot be calculated.

Lie algebras and loop quantum gravity

Mathematically, LQG is local gauge theory of the self-dual subgroup of the EXPANDED Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact group SO(3) or SU(2) as gauge group, as opposed to the non-compact groups SO(3,1) or SL(2.C). The compactness of the Lie group avoids some thus-far unsolved difficulties in the quantization of gauge theories of noncompact lie groups, and is responsible for the discreteness of the area and volume spectra. The theory involving the Immirzi parameter is necessary to resolve an ambiguity in the process of complexification. These are some of the many ways in which different quantizations of the same classical theory can result in inequivalent quantum theories, or even in the impossibility to carry quantization through.

It is not possible to distinguish between SO(3) and SU(2) or between SO(3,1) and SL(2,C) at this level: the respective Lie algebras are the same. All four groups have the same complexified Lie algebra and these subtleties are usually ignored in elementary particle physics. The physical interpretation of the Lie algebra is that of infinitesimally small group transformations, and gauge bosons (such as the graviton) are Lie algebra representations, not Lie group representations. This means for the Lorentz group that, for sufficiently small velocity parameters, all four complexified Lie groups are indistinguishable in the absence of matter fields.

Adding to the complexity, it can be shown that a positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group. At the level of the Lie algebra, this corresponds to what is called q-deforming the Lie algebra, and the parameter q is related to the value of the cosmological constant. The effect of replacing a Lie algebra by a q-deformed version is that the series of its representations is truncated. In the case of the rotation group, instead of having representations labelled by all half-integral spins, all representations with total spin j less than some constant remain.

It is possible to formulate LQG in terms of q-deformed Lie algebras instead of ordinary Lie algebras, and in the case of the Lorentz group, the result would, again, be indistinguishable for sufficiently small velocity parameters.

Spin networks loop quantum gravity

In the spin-foam formalism, the Barrett–Crane model, which was for a while the most promising state-sum model of 4D Lorentzian quantum gravity, was based on representations of the noncompact groups SO(3,1) or SL(2,C), so the spin foam faces (and hence the spin network edges) were labelled by positive real numbers as opposed to the half-integer labels of SU(2) spin networks.

These and other considerations, including difficulties interpreting what it would mean to apply a Lorentz transformation to a spin network state, led Lee Smolin and others to suggest that spin network states must break Lorentz invariance. Smolin and Joao Magueijo then went on to study doubly special relativity, in which not only there is a constant velocity c but also a constant distance l. They showed that there are nonlinear representations of the Lorentz Lie algebra with these properties (the usual Lorentz group being obtained from a linear representation). Doubly special relativity predicts deviations from the special relativity dispersion relation at large energies (corresponding to small wavelengths of the order of the constant length l in the doubly special theory). Giovanni Amelino-Camelia then proposed that the mystery of ultra-high-energy cosmic rays might be solved by assuming such violations of the special-relativity dispersion relation for photons. No confirmation has yet been found, and this idea is still hypothetical.

Phenomenological (hence, not specific to LQG) constraints on anomalous dispersion relations can be obtained by considering a variety of astrophysical experimental data, of which high-energy cosmic rays are one part.

Related Research Articles

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Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, such as neutron stars, as well as in the early stages of the universe moments after the Big Bang.

Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based directly on Albert Einstein's geometric formulation rather than the treatment of gravity as a mysterious mechanism (force). As a theory, LQG postulates that the structure of space and time is composed of finite loops woven into an extremely fine fabric or network. These networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale on the order of a Planck length, approximately 10⁻³⁵ meters, and smaller scales are meaningless. Consequently, not just matter, but space itself, prefers an atomic structure.

<span class="mw-page-title-main">Lee Smolin</span> American theoretical physicist (born 1955)

Lee Smolin is an American theoretical physicist, a faculty member at the Perimeter Institute for Theoretical Physics, an adjunct professor of physics at the University of Waterloo, and a member of the graduate faculty of the philosophy department at the University of Toronto. Smolin's 2006 book The Trouble with Physics criticized string theory as a viable scientific theory. He has made contributions to quantum gravity theory, in particular the approach known as loop quantum gravity. He advocates that the two primary approaches to quantum gravity, loop quantum gravity and string theory, can be reconciled as different aspects of the same underlying theory. He also advocates an alternative view on space and time that he calls temporal naturalism. His research interests also include cosmology, elementary particle theory, the foundations of quantum mechanics, and theoretical biology.

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In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles whose interactions are described by a gauge theory interact with each other by the exchange of gauge bosons, usually as virtual particles.

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References

↑ "So, how did everything start?". European Space Agency. 2003-06-06. Retrieved 2023-10-24.

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[1] "So, how did everything start?". European Space Agency. 2003-06-06. Retrieved 2023-10-24.

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