Pis’ma v ZhETF, vol. 109, iss. 6, pp. 369 - 370

March 25

Tetrads and q-theory

F. R. Klinkhamer⁺¹⁾, G. E. Volovik^∗×1)

⁺Institute for Theoretical Physics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany

^∗Low Temperature Laboratory, Aalto University, FI-00076 Aalto, Finland

^×Landau Institute for Theoretical Physics Russian Academy of Sciences, 119334 Moscow, Russia

Submitted 21 December 2018

Resubmitted 14 January 2019

Accepted 21

January 2019

DOI: 10.1134/S0370274X19060043

As the microscopic structure of the deep relativistic

classes of realizations of the q-variable (see Sec. 1 in [4]

quantum vacuum is unknown, a phenomenological ap-

for a general discussion of quantum-dissipative effects

proach (q-theory) has been proposed to describe the vac-

and [5] for a sample calculation).

uum degrees of freedom and the dynamics of the vacuum

Up till now, our discussions of q-theory have pri-

energy after the Big Bang. The original q-theory was

marily used the nonlinear theory of a four-form field

based on a four-form field strength from a three-form

strength from a three-form gauge potential (the lin-

gauge potential. However, this realization of q-theory,

ear theory of the vacuum energy in terms of the four-

just as others suggested so far, is rather artificial and

form field strength has been considered by Hawking [6],

does not take into account the fermionic nature of the

in particular). However, the four-form field strength,

vacuum. We now propose a more physical realization of

though useful for the construction of the general phe-

the q-variable. In this approach, we assume that the vac-

nomenological equations for the quantum vacuum, is

uum has the properties of a plastic (malleable) fermionic

rather abstract. The physical origin of such a field is not

crystalline medium. The new approach unites general

clear. In the new realization, the corresponding vacuum

relativity and fermionic microscopic (trans-Planckian)

variable q is expressed in terms of both the gravitational

degrees of freedom, as the approach involves both the

tetrad and the elasticity tetrad of the underlying crystal.

tetrad of standard gravity and the elasticity tetrad of

This realization in terms of tetrad fields is more appro-

the hypothetical vacuum crystal. This approach also al-

priate for the quantum theory of the fermionic vacuum

lows for the description of possible topological phases of

of our Universe than the realization with a bosonic four-

the quantum vacuum.

form field strength.

The q-theory framework [1, 2] provides a general

Throughout, we use natural units with c = ℏ = 1

phenomenological approach to the dynamics of vacuum

and take the metric signature (- + ++).

energy, which may be useful for the resolution of prob-

The tetrad formalism of torsion-less gravity is given

lems related to the cosmological constant in the Ein-

by the following equations:

stein equation (a brief review of q-theory appears in

g_µν = η_ab e^aµ e^bν, ∇_µ g^µν = 0,

(1a)

Appendix A of [3]). The advantage of q-theory is that,

at the classical level, the field equations of the theory

D_µ e^aν ≡ ∇_µ e^aν + ω^aµb e^bν = 0,

(1b)

essentially do not depend on the detailed microscopic

where ∇_µ is the conventional covariant derivative of gen-

(trans-Planckian) origin of the q-field. In the classical

eral relativity and ω^aµb the spin connection,

limit, the field equations of q-theory (i.e., the equation

for the microscopic variable q and the modified Einstein

ω^aµb = e^aν ∇_µ e^νb.

(2)

equation for the metric) are universal.

We interpret the vacuum as a plastic (malleable)

The q-theory approach to the cosmological constant

fermionic crystalline medium. At each point of space-

problem aims to describe the decay of the vacuum en-

time, we have a local system of four deformed crystal-

ergy density from an initial Planck-scale value to the

lographic manifolds of constant phase X^a(x) = 2πn^a,

present value of the cosmological constant. However, the

for n^a ∈ Z with a = 0, 1, 2, 3. In addition to the con-

correct description of this decay requires the quantum

ventional tetrad e^aµ of gravity, we then introduce the

version of q-theory, which may be different for different

following elasticity tetrad E^aµ (cf. [7-9]):

¹⁾e-mail: frans.klinkhamer@kit.edu; volovik@ltl.tkk.fi

E^aµ(x) = D_µ X^a(x),

(3)

Письма в ЖЭТФ том 109 вып. 5 - 6

2019

369

370

F. R. Klinkhamer, G. E. Volovik

where both indices a and µ take values from the set

ρ_V (q₀) = 0,

(13a)

{0, 1, 2, 3}. Invariance under the local SO(1, 3) group

[dρV (q)]

of rotations is implemented by defining

= 0,

(13b)

q=q0

D_µ X^a ≡ ∇_µ X^a + ω^aµb X^b = ∂_µ X^a + ω^aµb X^b.

(4)

[d2 ρV (q)]

> 0.

(13c)

dq²

Let us assume that the vacuum energy density ǫ(q)

q=q0

in the action depends on the following type of q-field:

Equations

(13a) and

(13b) result from the self-

adjustment of the conserved vacuum variable q, as

q(x) =

e^µa(x)E^aµ(x).

(5)

follows from the Gibbs-Duhem relation for an isolated

self-sustained system without external pressure; Eq.

The action in its simplest form is then given by

(13c) corresponds to positive isothermal compressibility

of the vacuum.

∫

(

)

To summarize, we have obtained with (5) one further

d⁴xe

+ ǫ(q)

(6)

R⁴

16πG_N

realization of the q-variable, in addition to the four-form

realization [1] and the brane realization [2]. The advan-

where R is the Ricci curvature scalar and e the tetrad

tage of this new realization is that it has a more di-

determinant. Variation of Eq. (6) over e^µa gives the Ein-

rect physical origin. The quantum version of q-theory

stein equation [10],

is sensitive to the particular realization of the q-field.

Assuming q-theory to be relevant, the comparison with

R_µν -

g_µν R = 8πG_N ρ_V (q)g_µν,

(7)

experiment may then provide information on the de-

tailed structure of the fermionic quantum vacuum and,

where ρ_V (q) will be discussed shortly, and variation over

in particular, on the types of quantum anomalies. The

X^a gives the following differential equation for q (which

fermionic crystalline model of the vacuum is one of the

is both a coordinate scalar and a Lorentz scalar):

possible structures of the deep fermionic vacuum, dis-

tinct from a structure described the abstract four- form

( dǫ(q))

field strength. This new fermionic structure gives, for ex-

∂_µ

= 0,

(8)

ample, rise to new types of quantum anomalies, where

elasticity tetrads E with dimensions of inverse length or

where (1b) has been used. The vacuum energy density

inverse time are mixed with gauge and gravity fields [9].

ρ_V (q), which enters the Einstein equation (7) through a

The work of G. E. Volovik has been supported by the

cosmological-constant-type term, is given by

European Research Council (ERC) under the European

dǫ(q)

Union’s Horizon

2020

research and innovation pro-

ρ_V (q) ≡ ǫ(q) - q

(9)

gramme (Grant Agreement #694248).

Full text of the paper is published in JETP Letters

with an extra term -q dǫ/dq.

journal. DOI: 10.1134/S0021364019060031

Equation (8) for q has the following general solution:

1. F. R. Klinkhamer and G. E. Volovik, Phys. Rev. D 77,

dǫ(q)

085015 (2008).

= µ = constant,

(10)

2. F. R. Klinkhamer and G. E. Volovik, JETP Lett. 103,

627 (2016).

with the arbitrary constant µ interpreted as a “chemi-

3. F. R. Klinkhamer and G. E. Volovik, J. Phys. Conf. Ser.

cal potential” in [1], and the gravitating vacuum energy

314, 012004 (2011).

density (9) becomes ρ_V (q) = ǫ(q) - µ q.

4. F. R. Klinkhamer, M. Savelainen, and G. E. Volovik,

JETP 125, 268 (2017).

The quantum vacuum in perfect equilibrium has a

5. F. R. Klinkhamer and G. E. Volovik, Mod. Phys. Lett.

constant nonzero value of the q-field,

A 31, 1650160 (2016).

6. S. W. Hawking, Phys. Lett. B 134, 403 (1984).

q(x) = q₀ = constant,

(11)

7. I. E. Dzyaloshinskii and G. E. Volovick, Ann. Phys. 125,

67 (1980).

which gives a particular value µ₀ for µ in (10),

8. J. Nissinen and G. E. Volovik, JETP 127, 948 (2018).

]

9. J. Nissinen and G.E. Volovik, arXiv:1812.03175.

[ dǫ(q)

µ₀ =

(12)

10. S. Weinberg, Gravitation and Cosmology: Principles and

q=q0

Applications of the General Theory of Relativity, John

Wiley and Sons, N.Y. (1972).

In addition, there are the equilibrium conditions:

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2019