Abstract
We consider the possibility of the scenario in which the P, T and Lorentz symmetry of the relativistic quantum vacuum are all the combined symmetries. These symmetries emerge as a result of the symmetry breaking of the more fundamental P, T and Lorentz symmetries of the original vacuum, which is invariant under separate groups of the coordinate transformations and spin rotations. The condensed matter vacua (ground states) suggest two possible scenarios of the origin of the combined Lorentz symmetry, and both are realized in the superfluid phases of liquid ^3He: the ^3He-A scenario and the ^3He-B scenario. In these scenarios, the gravitational tetrads are considered as the order parameter of the symmetry breaking in the quantum vacuum. The ^3He-B scenarios applied to the Minkowski vacuum lead to the continuous degeneracy of the Minkowski vacuum with respect to the O(3, 1) spin rotations. The symmetry breaking leads to the corresponding topological objects, which appear due to the nontrivial topology of the manifold of the degenerate Minkowski vacua, such as torsion strings. The fourfold degeneracy of the Minkowski vacuum with respect to discrete P and T symmetries suggests that the Weyl fermions are described by four different tetrad fields: the tetrad for the left-handed fermions, the tetrad for the right-handed fermions, and the tetrads for their antiparticles. This may lead to the gravity with several metric fields, so that the parity violation may lead to the breaking of equivalence principle. Finally, we considered the application of the gravitational tetrads for the solution of the cosmological constant problem.
Highlights
Topological superfluid phases of liquid 3 He provide many connections with the Standard Model of particle physics and gravity [1, 2]
The order parameter eia breaks both rotational symmetries of the original Hamiltonian, since it transforms under SO(3)c group of the coordinate rotations and under SO(3)s group of the spin rotations
We considered two scenarios of the formation of the combined symmetry describing the tetrad gravity interacting with fermions: the scenario of emergent symmetry suggested by analogy with the chiral superfluid 3He-A and the scenario of symmetry breaking, which is suggested by the symmetry breaking pattern in topological superfluid 3He-B
Summary
Topological superfluid phases of liquid 3 He provide many connections with the Standard Model of particle physics and gravity [1, 2]. The fermionic action lacks the separate symmetries, such as symmetry with respect to the only coordinate transformations Lc , but is invariant under the combined operation, L = LcLs , i.e., when the space rotations are accompanied to the spin rotations of the Dirac or Weyl spinors. In the B-phase, this symmetry is reduced to the diagonal group G → H = SO(3)J of the combined rotations Such symmetry breaking is known as the broken relative symmetry [6], the vacuum becomes degenerate with respect to the separate coordinate or spin rotations. Two independent Lorentz groups of the coordinate and spin rotations are broken to the diagonal subgroup of the degenerate Minkowski vacuum. The broken discrete symmetries and the corresponding fourfold degeneracy of the Minkowski vacuum suggest that the Weyl fermions can be described by four tetrad fields: each for left-handed and right-handed fermions and for their antiparticles.
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