Pis’ma v ZhETF, vol. 111, iss. 7, pp. 441 - 442

April 10

On dimension of tetrads in effective gravity

G. E. Volovik¹⁾

Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 AALTO, Finland

Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

Submitted 29 February 2020

Resubmitted 10 March 2020

Accepted 11

March 2020

DOI: 10.31857/S0370274X20070024

There are several scenarios of emergent gravity.

a vanishing determinant of the metric [15, 16], which

Gravity may emerge in the vicinity of the topologically

would correspond of the vacuum state with unbroken

stable Weyl point [1-5]; the analog of curved spacetime

symmetry, i.e., with zero tetrad field, e^Aµ = 0. On the

emerges in hydrodynamics with the so-called acoustic

other hand, the Big Bang can be considered as a symme-

metric for the propagating sound waves [6]; etc. Here

try breaking phase transition L_L×L_S → L, at which the

we consider two very different scenarios, which how-

symmetry between the spacetime with e > 0 and anti-

ever have unusual common property: the tetrad fields

spacetime with e < 0 is spontaneously broken, where

in these theories have dimension of inverse length. As

e is the tetrad determinant. Correspondingly, in super-

a result all the physical quantities which obey diffeo-

fluid³He the formation of the p-wave order parame-

morphism invariance are dimensionless. This was first

ter spontaneously breaks the symmetry under coordi-

noticed by Diakonov [7] and Vladimirov and Diakonov

nate transformation r → -r. The VD scenario has also

(VD) [8, 9] in the scenario, where tetrad fields emerge

the connection to the chiral³He-A phase: in both sys-

as bilinear combinations of the fermionic fields. Tetrads

tems the topologically protected Weyl fermions emerge,

with dimension of inverse length emerge also in the

which move in the effective tetrad field [5].

model of the superplastic vacuum [10, 11].

According to Eq. (1), the frame field e^Aµ transforms

In the theory by VD [7-9] the tetrads are composite

as a derivative and thus has the dimension of inverse

fields, which emerge as the bilinear combinations of the

length, [e^Aµ] = 1/[l] (it is assumed that ψ is scalar un-

fermionic fields:

der diffeomorphisms) [7, 8]. For Weyl or massless Dirac

fermions one ∫as the conventional action:

e^Aµ = i ψ^†γ^A∇_µψ + ∇_µψ^†γ^Aψ .

(1)

(

)

S = d⁴x|e|e^Aµ

ψ^†γ^A∇_µψ + H.c.

(2)

This construction is similar to what happens in the spin-

triplet p-wave superfluids in the³He-B phase [12]. In

The action (2) expressed in terms of the VD tetrads is

the VD scenario two separate Lorentz groups of coor-

dimensionless, since [e] = [l]^-4, [e^Aµ] = [l] and [ψ] = 1.

dinate and spin rotations are spontaneously broken to

The elasticity tetrads describe elasticity theory [10,

the combined Lorentz symmetry group, L_L × L_S → L.

11, 17, 18]. In conventional crystals they are gradients

In the same manner in³He-B the symmetries under

of the three U(1) phase fields X^A, A = 1, 2, 3,

three-dimensional rotations in orbital and spin spaces

e Aµ(x) = ∂_µX^A(x).

(3)

are broken to the symmetry group of combined rota-

tions, SO(3)_L × SO(3)_S → SO(3)_J .

The surfaces of constant phases, X^A(x)

= 2πn^A,

Formation of tetrads breaks both the symmetries un-

describe the system of the deformed crystallographic

der discrete coordinate transformations P_L = (r → -r)

planes. Being the derivatives, elasticity tetrads have also

and T_L = (t → -t), and the discrete symmetries in

canonical dimensions of inverse length. This allows us to

spin space, P_S and T_S . The symmetry breaking scheme

extend the application of the topological anomalies. The

P_L × P_S → P and T_L × T_S → T leaves the combined

Chern-Simons term describing the 3 + 1 quantum Hall

parity P and the combined time reversal symmetry T .

effect becomes dimensionless. As a result, the prefactor

The VD symmetry breaking mechanism can be im-

of term is given by the integer momentum-space topo-

portant for the consideration of the Big Bang scenario,

logical invariants in the same manner as in the case of

in which the gravitational tetrads change sign across the

2+1 dimension.

The elasticity tetrads can be used as the gravita-

singularity, e^Aµ(τ, x) = -e^Aµ(-τ, x) [13, 14]. The singu-

tional tetrads for the construction of gravity in the

larity can be avoided by formation of the bubble with

model of the

3+1 vacuum as a plastic (malleable)

¹⁾e-mail: grigori.volovik@aalto.fi

fermionic crystalline medium with A = 0, 1, 2, 3 [19, 20].

Письма в ЖЭТФ том 111 вып. 7 - 8

2020

441

442

G. E. Volovik

In plastic vacuum all physical quantities become dimen-

momentum-space invariant. The relativistic example is

sionless [11]. Such vacuum can be arbitrarily deformed,

the chiral anomaly in terms of torsion fields [23, 24]. For

and thus the equilibrium microscopic length scale (such

the torsion and curvature in terms of the conventional

as Planck scale) is absent. All distances are measured in

tetrads, the gravitational Nieh-Yan anomaly equation

terms of the integer positions of nodes of plastic crystal,

for the non-conser(ation of the axial current

)

and the Newton constant, the scalar curvature R, the

∂_µj^µ5 = λ²

T^A ∧ T_A - e^A ∧ e^B ∧ R_AB

(6)

cosmological constant Λ, and particle masses M become

contains the nonuniversal prefactor - the ultraviolet cut-

dimensionless [11].

off parameter λ with dimension [λ] = 1/[l], which may

The same is for VD gravity, where “all world scalars

depend on the spacetime coordinates, explicitly violat-

are dimensionless, be it the scalar curvature R, the in-

ing the topology. In terms of VD tetrads, the prefactor λ

terval ds, the fermion field ψ, or any diffeomorphism-

becomes dimensionless, [λ] = 1, which properly reflects

invariant action term” [8]. Example is the mass term:

the topology of the quantum vacuum.

∫

This work has been supported by the European

S = d⁴x|e|Mψ^†ψ,

(4)

Research Council (ERC) under the European Union’s

Horizon

2020

research and innovation programme

[e] = [l]^-4, [ψ] = 1 and [M] = 1. For bosonic scalar field

∫

(Grant Agreement # 694248).

S = d⁴x√-g(gµν ∇µΦ∇ν Φ + M2Φ2) ,

(5)

Full text of the paper is published in JETP Letters

journal. DOI: 10.1134/S0021364020070024

one has [g^µν ] = [l]², [√-g] = [l]-4, [Φ] = 1 and [M] = 1.

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2020