Lorentz invariance in loop quantum gravity
Encyclopedia
Loop quantum gravity
(LQG) is a quantization
of a classical Lagrangian
field theory. It is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity
with torsion
. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just like it is broken in general relativity
(unless one is dealing with Minkowski spacetime, which is one particular solution of the Einstein field equations). On the other hand, there has been much talk about possible local and global violations of Lorentz invariance beyond those expected in straightforward general relativity. Of interest in this connection would be to see whether the LQG analogue of Minkowski spacetime breaks or preserves global Lorentz invariance, and Carlo Rovelli
and coworkers have recently been investigating the Minkowski state of LQG using spin-foam techniques.
These questions will all remain open as long as the classical limit
s of various LQG models (see below for the sources of variation) cannot be calculated.
Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified 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.
One can't 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. In fact, all four groups have the same complexified Lie algebra, which makes matters even more confusing (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. What this means for the Lorentz group is that, for sufficiently small velocity parameters, all four complexified Lie groups are indistinguishable in the absence of matter fields.
To make matters more complicated, 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, one is left with all representations with total spin j less than some constant). It is entirely 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.
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. Lee 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.
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 but one part. Current observations are already able to place exceedingly stringent constraints on these phenomenological parameters.
Loop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
(LQG) is a quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of a classical Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
field theory. It is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
with torsion
Torsion
The word torsion may refer to the following:*In geometry:** Torsion of a curve** Torsion tensor in differential geometry** The closely related concepts of Reidemeister torsion and analytic torsion ** Whitehead torsion*In algebra:** Torsion ** Tor functor* In medicine:** Ovarian...
. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just like it is broken in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
(unless one is dealing with Minkowski spacetime, which is one particular solution of the Einstein field equations). On the other hand, there has been much talk about possible local and global violations of Lorentz invariance beyond those expected in straightforward general relativity. Of interest in this connection would be to see whether the LQG analogue of Minkowski spacetime breaks or preserves global Lorentz invariance, and Carlo Rovelli
Carlo Rovelli
Carlo Rovelli is an Italian physicist who has worked in Italy, the USA, and France. His work is mainly in the field of quantum gravity. He is among the founders of the Loop Quantum Gravity theory.-Life and career:...
and coworkers have recently been investigating the Minkowski state of LQG using spin-foam techniques.
These questions will all remain open as long as the classical limit
Classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior...
s of various LQG models (see below for the sources of variation) cannot be calculated.
Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
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
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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
Immirzi parameter
The Immirzi parameter is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units...
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.
One can't 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. In fact, all four groups have the same complexified Lie algebra, which makes matters even more confusing (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
Graviton
In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...
) are Lie algebra representations, not Lie group representations. What this means for the Lorentz group is that, for sufficiently small velocity parameters, all four complexified Lie groups are indistinguishable in the absence of matter fields.
To make matters more complicated, it can be shown that a positive cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
can be realized in LQG by replacing the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
with the corresponding quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
. 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, one is left with all representations with total spin j less than some constant). It is entirely 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.
In the spin-foam formalism, the Barrett-Crane model
Barrett-Crane model
The Barrett–Crane model is a model in quantum gravity which was defined using the Plebanski action.The B field in the action is supposed to be a so-valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term...
, 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
Lee Smolin
Lee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M...
and others to suggest that spin network states must break Lorentz invariance. Lee Smolin
Lee Smolin
Lee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M...
and Joao Magueijo
João Magueijo
João Magueijo is a Portuguese cosmologist and professor in Theoretical Physics at Imperial College London. He is a pioneer of the varying speed of light theory.- Career :...
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
Dispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....
at large energies (corresponding to small wavelengths of the order of the constant length l in the doubly special theory). Giovanni Amelino-Camelia
Giovanni Amelino-Camelia
Giovanni Amelino-Camelia is an Italian physicist of the University of Rome La Sapienza who works on quantum gravity.He is the first proposer of Doubly special relativity that is the idea of introducing the Planck length in physics as an observer-independent quantity, obtaining a relativistic theory...
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.
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 but one part. Current observations are already able to place exceedingly stringent constraints on these phenomenological parameters.