ЖЭТФ, 2021, том 159, вып. 4, стр. 815-821

DIMENSIONLESS PHYSICS

G. E. Volovik^*

Low Temperature Laboratory, Aalto University

P. O. Box 15100, FI-00076 Aalto, Finland

Landau Institute for Theoretical Physics, Russian Academy of Sciences

142432, Chernogolovka, Moscow Region, Russia

Received June 30, 2020,

revised version August 11, 2020

Accepted for publication August 12, 2020

Contribution for the JETP special issue in honor of I. E. Dzyaloshinskii’s 90th birthday

DOI: 10.31857/S0044451021040246

The first one is the tetrad gravity, where the tetrad

fields emerge as bilinear combinations of the fermionic

fields by symmetry breaking. This scenario has been in-

It is not excluded that the Standard Model of parti-

vestigated by Diakonov [9], Vladimirov and Diakonov

cle physics together with general relativity are effective

[10, 11], and Obukhov and Hehl [12]. The analog

theories which emerge at low energy for the collective

of Diakonov-Vladimirov (DV) scenario takes place in

modes of the extreme ultraviolet [1]. Sakharov grav-

topological superfluid³He-B [13].

ity [2], which emerges in the fermionic vacuum, pro-

vides the characteristic example. The scenario, where

The other one is the analog of gravity in the elastici-

all the known symmetries in our Universe emerge on

ty theory of crystals [14-20], where the elastic deforma-

the macroscopic scale, but disappear in the highly

tions are described in terms of the tetrads of elasticity

trans-Planckian microscopic regime, takes place in dif-

[16]. In principle, this analogy can be extended to the

ferent many-body condensed matter systems. For ex-

real gravity, if the quantum vacuum is considered as

ample, the analogs of Lorentz invariance and the curved

a plastic (malleable) fermionic crystalline medium and

spacetime are developed for some low energy fermionic

the elasticity tetrads become the gravitational tetrads

and bosonic modes [3], but these phenomena disap-

[21, 22]. The condensed matter analog of such vacuum

pear at large energy, where the microscopic degrees of

is the quantum crystal with fermionic quasiparticles,

freedom intervene (the analog of the trans-Planckian

such as vacancies [23-25].

degrees of freedom). The condensed matter systems

The common property of these two approaches to

with topologically stable Weyl points in the fermionic

spectrum, such as Weyl semimetals and the chiral su-

quantum gravity is that the tetrad fields in both theo-

ries have dimension of inverse length. As a result, most

perfluid phase of liquid³He, demonstrate simultaneous

emergence of chiral fermions, gauge bosons, and tetrad

of the physical quantities which obey diffeomorphism

gravity [4-8], which do not survive on the high-energy

invariance become dimensionless [11, 26-28]. Since the

two very different scenarios lead to the same phe-

atomic level.

nomenon, it is natural to suggest that the gravity in

We do not know the structure of the trans-Planc-

our universe also follows this common rule. Here we

kian world, but we can try different possible scenarios

consider some consequences which come from this rule.

of emergent physics and search for the common prop-

erties in the low energy corner. Here we consider two

Let us consider the theory of the crystal elastic-

scenarios of emergent gravity, which are very different,

ity using approach of Ref. [16]. The deformed crystal

but have the important common property.

structure can be described as a system of three crystal-

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

n^a ∈ Z with a = 1, 2, 3. The intersection of the sur-

^* E-mail: volovik@boojum.hut.fi

faces

815

G. E. Volovik

ЖЭТФ, том 159, вып. 4, 2021

X¹(r, t) = 2πn¹, X²(r, t) = 2πn²,

The shift of the dimensions of the physical quanti-

(1)

X³(r, t) = 2πn³

ties leads to the new properties of the quantum vacua

and also topological insulators, which allows extending

are the nodes of the deformed crystal lattice. For the

the application of the topological anomalies. For exam-

undeformed crystal, X^a(r, t) = K^a · r, where K^a are

ple, the Chern-Simons term describing the 3+1 intrin-

the (primitive) reciprocal lattice vectors. The defor-

sic quantum Hall effect becomes dimensionless. As a

mations of the crystal can be described in terms of the

result, the prefactor of this term is given by the integer

elasticity tetrads, the gradients of the phase functions:

momentum-space topological invariants as in the case

of 2 + 1 dimension [26-28]. The shift is also important

E ai(x) = ∂_iX^a(x).

(2)

for the Nieh-Yan anomaly [28].

In the DV theory [9-12], the tetrads are composite

In the absence of dislocations, E^ai(x) is an exact dif-

fields emerging as bilinear combinations of fermionic

ferential:

fields. Tetrads appear as the order parameter of the

∂_kE^al(x) - ∂_lE^ak(x) = 0 .

(3)

symmetry breaking transition (see also Ref. [29]):

In the presence of the topological defects - dislocations,

e^aμ = i ψ^†Γ^a∇_μψ - ∇_μψ^†Γ^aψ

(6)

the density of dislocations plays the role of torsion:

The corresponding symmetry breaking scheme is L_L ×

T^akl = (∂_kE^al - ∂_lE^ak).

(4)

× L_S → LL+S, where L_L and L_S are two separate

symmetries under Lorentz rotations of the coordinate

Such construction can be extended to the 3+1 quan-

and spin space, respectively. These two symmetries are

tum vacuum with a = 0, 1, 2, 3, assuming that the vac-

broken to the diagonal subgroup — the Lorentz group

uum looks like the plastic crystalline medium. In this

of the combined rotations in two spaces, LL+S. In addi-

model the elasticity tetrads E^aμ become the gravita-

tion, this order parameter breaks the P T symmetry, see

tional tetrads [21, 22]. The deformed vacuum crystal

also Ref. [30]. The similar scheme of symmetry brea-

with dislocations describes the curved geometry of the

king of three-dimensional rotations in the orbital and

teleparallel Weitzenböck gravity with vanishing curva-

spin spaces takes place in the superfluid³He-B [13,31]:

ture and nonzero torsion. On the macroscopic coarse

grained level, where the separate dislocations are not

SO(3)_L × SO(3)_S → SO(3)S+L.

resolved, the torsion field T^aμν can be considered as a

continuous function of coordinates. The metric g_μν

According to Eq. (6), the frame field e^aμ transforms

originates from the elasticity tetrads:

as a derivative in the same manner as the elasticity

tetrads. That is why it has the dimension of inverse

g_μν = η_abE^aμE^bν ,

(5)

length, [e^aμ] = 1/[l], i. e., its dimension is shifted from

= 1, see Table 2. In such vacua, it is

d = 0 to d_DV

where η_ab = (-, +, +, +).

natural to assume that the fermionic field ψ as well as

The important property of the elasticity tetrads is

the bosonic fields Φ are scalars under diffeomorphisms

that being the derivatives, they have the canonical di-

[9, 10], i. e., their dimensions are shifted from d = 3/2

mensions of inverse length, [E^aμ] = [l]-1. Correspond-

and d = 1 to dDV = 0, see Tables 2 and 3, respec-

ingly, the metric has dimension [g_μν ] = 1/[l]², while the

tively. For Weyl or massless Dirac fermions one has

interval is dimensionless: ds² = -g_μν dx^μdx^ν , [Δs] = 1.

the conventional action:

The distance between the two nodes of the deformed

∫

(

)

crystal is determined by the integer number of crys-

S = d⁴x|e|e^aμ

ψ^†Γ^a∇_μψ + H.c.

tal surfaces between the points of the grid and thus

does not depend on the length scale. In such quantum

This action expressed in terms of the DV tetrads re-

vacua, the size of the unit cell is not fixed and can be

mains dimensionless, since [e] = [l]-4, [e^aμ] = [l], and

arbitrary.

[ψ] = 1, see Table 2. This suggests that the DV dimen-

Let us introduce notation d for conventional dimen-

sion of tetrads is natural, which is also supported by

sion of the physical quantities and the notation d_DV for

the elasticity tetrads.

the shifted dimension of the same quantities. The shift

The nontrivial dimension of the metric suggests that

of dimensions means that [l]^-d → [l]-dDV in the DV

metric is not the quantity, which describes the space-

approach. For the interval ds, the conventional dimen-

time, but the quantity, which determines the dynamics

sion d = -1 and the shifted dimension dDV = 0, see

of effective low energy fields in the background of mic-

Table 1. For the torsion field d = 1, and d_DV = 2.

roscopic quantum vacuum.

816

ЖЭТФ, том 159, вып. 4, 2021

Dimensionless physics

The shifts of dimensions are shown in Tables 1, 2,

energy scale [38,39] and the neutrino mass scale [40] en-

and 3 correspondingly for gravity, fermions, and scalar

ter, ΛG² ∼ M8W G⁴ and ΛG² ∼ M4nG², respectively.

∫

fields. Many quantities, which obey diffeomorphism

The spacetime volume V =

d⁴x√-g is dimen-

invariance, become dimensionless. The action is di-

sionless, [V ] = 1, and may have quantized values. Then

mensionless and remains dimensionless in the DV di-

Λ as the corresponding Lagrange multiplier may have

mensions, since the action is a diffeomorphism invari-

universal quantized values with Λ = 0 in the equilib-

ant. Another example of the diffeomorphism invariant

rium Minkowski vacuum.

quantity is the rest mass M of particles. In the case of

The dimensionless Lagrange multiplier appears also

mass, the dimension is shifted from d = 1 to d_DV = 0.

in the q-theory of the quantum vacuum [41], if the

That the DV dimension of mass is [M] = 1 can be

q-theory is based on the 4-form gauge field introduced

seen from the classical equation for the particle spec-

by Hawking for phenomenological description of the

trum: g^μν p_μp_ν = M². According to Table 1, M² has

quantum vacuum [42],

dimension d_DV = -2 + 1 + 1 = 0, i. e., [M] = 1, and

q² = F^μναβF_μναβ.

the action and the mass terms in fermionic and bosonic

actions are dimensionless:

In the DV units the vacuum variable q and the La-

∫

grange multiplier μ_q (the corresponding chemical po-

S = M ds,

(7)

tential of the conserved quantity) are dimensionless,

see Table 3. If μ_q is fundamental, it becomes the gen-

∫

eral characteristics of the quantum vacuum. While

S = d⁴x|e|Mψ^†ψ ,

(8)

the variable q determines the variable vacuum energy

Λ(q) = ϵ(q)-μ_qq, the universal chemical potential pro-

vides the nullification of Λ in the Minkowski vacuum.

∫

At this value of μ_q, all the initial states (even those

S= d⁴x√-g(gμν∇μΦ∇νΦ + M2Φ2) ,

(9)

with the Planck scale Λ) finally relax to Minkowski

vacuum with Λ = 0 [41], thus providing the solution of

∫

the cosmological constant problems.

S =

d⁴x√-g(Fμν Fμν + M2gμν AμAν ) .

(10)

In the DV approach, mass and energy have differ-

ent dimensions. While mass is dimensionless, [M] = 1,

This follows from the DV dimensions

the energy has dimension of frequency, [E] = [ω] =

=[√g00] = 1/[l]. Correspondingly, the temperature is

[e] = [√-g] = [l]-4, [Φ] = [ψ] = 1,

[g^μν ] = [l]-2,

dimensionless, [T ] = 1, while the constant temperature,

√

which enters the Tolman law [43], T (r)

-g₀₀(r) =

[ds] = [M] = 1,

[A_μ] = [l]-1.

= T_Tolman, has dimension of frequency, [T_Tolman] =

Since the scalar curvature in general relativity is

= [ω] = [√g00] = 1/[l], see Table 1. Tolman tempe-

diffeomorphism invariant, it is dimensionless in the DV

rature is the integration constant in equilibrium in a

approach, [R] = 1. Its dimension is shifted from d = 2

stationary spacetime [44].

to d_DV = 0. Other examples of the diffeomorphism in-

The Unruh temperature of the accelerated body is

variant quantities are the Newton constant G and the

T_U = a/2π [45], where a is covariant acceleration,

cosmological constant Λ in the Einstein-Hilbert action:

d²x^μ d²x^ν

∫

a²

=g_μν

√

S_GR =

d⁴x

-g R + d⁴x√-g Λ .

(11)

ds²

16πG

Since a is diffeomorphism-invariant, it is dimensionless

The dimensions of G and Λ are shifted from corre-

together with the Unruh temperature, [a] = [T_U ] = 1.

spondingly d = -2 and d = 4 to d_DV = 0, i. e., [Λ] =

The same is with the Hawking temperature of a black

= [G] = 1. These dimensionless quantitites are deter-

hole. For the Schwartzschild black hole with rest energy

mined by the ratio of the mass scales [32] or by the

M_BH, Bekenstein entropy S_BH, Hawking temperature

functions of scalar fields [33]. In principle, only the ra-

T_BH, and horizon area A_BH:

tio between the mass parameters makes sense [10]. In

a given case, only the combination ΛG² matters. Ac-

T_BH =

8πGM_BH

cording to Zeldovich [32], this combination is expressed

(12)

in terms of QCD mass scale: ΛG² ∼ Λ6QCDG³ (see also

A_BH

Refs. [34-37]). In the other approaches, the electroweak

S_BH = 4πGM2BH =

817

ЖЭТФ, вып. 4

G. E. Volovik

ЖЭТФ, том 159, вып. 4, 2021

Table 1. Dimension shifts for gravity

Table 2. Dimension shifts for fermions

General relativity

Dimension d

d_DV

Fermions

Dimension d d_DV

g^μν

-2

e^aμ

g_μν

e^μa

-1

√-g

d⁴x

√-g

-4

3/2

ds² = g_μν dx^μdx^ν

-2

ψψ

√

dA =

dS^μν dS_μν

-2

ψΓ^ae^μaD_μψ

T_a

∫

S=M

T_aT^a

∂_μ

λ2Nieh-Y an

p_μ

e^aμA_ν

F^μν

R=g^μνR_μν

ie^μae^νbψ(ΓaΓb-ΓbΓa)ψFμν

G_Newton

-2

QQQL

R/G_Newton

Λ_cosmological

Since

T_Hawking

T_Tolman

[S^μν ] = [l]² and

[S_μν ] = [S^μν ][g_μν ]² = 1/[l]²,

T_Tolman/

√g₀₀

one obtains [A] = 1, which supports the idea that the

area of the black hole horizon is quantized [46-48].

Table 3. Dimension shifts for scalar fields

Similar quantization may occur for the de Sit-

ter spacetime, which is the submanifold of Minkowski

spacetime in the 4 + 1 dimension:

Scalar/Vector

Dimension d

d_DV

g4+1Minkμνx^μx^ν = α².

g^μν∂_μΦ∂_νΦ

Since [g_μν ] = [l]-2, the parameter α is dimensionless

M²φ²

as well as the scalar curvature R = 12/α², i. e., in the

A_μ

DV dimensions [R] = [α] = 1. The dimensionless pa-

rameter α of the de Sitter spacetime emphasizes the

F_μν

unique symmetry of this spacetime and supports quan-

F_μνF^μν

tization of this parameter (see, e. g., [49]), which could

(F_μν F^μν )^k

be similar to the Bekenstein quantization of the black

F_μν

F^μν

hole area [46]. However, in case of the superplastic

F_μναβ

vacuum, the quantization of area can be very different

from the quantization in terms of the Planck area, be-

F^μναβ

-4

cause the elementary cell of the underlying lattice may

q² = F^μναβF_μναβ

have nothing to do with the Planck scale.

μ_q

Table 2 contains the operators with the mass dimen-

sions 3, 5, and 6. The non-renormalisable dimension 5

operator gives a contribution to the electron magnetic

All the quantities, that enter Eq. (12), are dimension-

moment [1]:

less in the DV approach, [T_BH ] = [S_BH ] = [M_BH ] =

G₅ = ie^μae^νb

ψ(Γ^aΓ^b - Γ^bΓ^a)ψF_μν ,

= [A_BH ] = [G] = 1. The area of the black hole is

dimensionless, because the covariant form of the scalar

and the non-renormalisable dimension 6 four-fermion

area element is

operator describes the baryon number violation:

√

dA =

dS^μν dS_μν .

G₆ = QQQL,

818

ЖЭТФ, том 159, вып. 4, 2021

Dimensionless physics

where L and Q are the lepton and quark doublets.

contains the nonuniversal prefactor — the ultraviolet

Since in the DV approach the mass is dimensionless,

cut-off parameter λ with dimension of inverse length,

these operators become dimensionless: their d_DV = 0.

[λ] = 1/[l]. Since λ may depend on coordinates, which

The prefactors in these terms are determined either

explicitly violates the topology, the Nie-Yan contribu-

by the ratio of the mass scales (“ultraviolet” and “in-

tion to the anomaly is still rather subtle (see recent

frared”) or by the functions of scalar fields. The same

literature [27, 61-65]). In the DV tetrads, the torsion

is with the 4k mass operator for k > 1 in Table 3:

in Eq. (4) has dimension [T^akl] = 1/[l]², and the prefac-

G4k = (F_μνF^μν)^k.

tor λ² becomes dimensionless, [λ] = 1, which suggests

In terms of the DV dimensionalities, the operators

that the prefactor is universal and is quantized.

with dDV = 4 are topological. The operators of the

The Chern-Simons term describing the 3+1 quan-

type G₄ = F_μνFμν are topological in both classes of

tum Hall effect can be extended to the

3+1+1

dimensions, since for them d = dDV = 4. They are

Wess-Zumino actions:

accompanied by the fundamental integer or fractional

∫

prefactors. There are also operators, which have orig-

SaabWZ =

d⁴xdτ ϵ^μναβγe^aγFaμνFbαβ .

(17)

8π²

inal dimension d = 4 but acquire dimension d_DV = 4

X⁵

in the DV approach. This means that they are not

∫

topological in conventional approach, but may become

S^abcNY ∝ d⁴xdτ ϵ^μναβγe^aγT^bμνT^cαβ .

(18)

topological in the DV dimensions.

X⁵

The former dimension d = 2 operator T_aT^a and the

∫

dimension d = 3 operator e^aμA_νFμν acquire dimension

S^abWZ ∝ d⁴xdτ ϵ^μναβγe^aγT^bμνF_αβ .

(19)

d_DV = 4 in the DV dimensionalities. As a result, they

become topological and their prefactors become the

In terms of DV tetrads, these dimensionless terms are

topological quantum numbers. The operator e^aμA_νFμν

universal and do not depend on the cut-off parameters.

determines the quantum Hall response in 3 + 1 topo-

In two scenarios of emergent gravity, the superplas-

logical insulators [26, 27, 50] described by the following

tic vacuum and the DV theory with bilinear tetrad

Chern-Simons topological term [27]:

field, the invariance under diffeomorphisms leads to the

dimensionless physics. In the DV theory, this invari-

S4D[A_μ] =

ance is assumed as fundamental. In the superplastic

∫

vacuum, the diffeomorphism invariance corresponds to

∑

N_a d⁴x E^aμϵ^μναβA_ν∂_αA_β .

(13)

the proposed invariance under arbitrary deformations

4π²

a=1

of the 4D vacuum crystal. In words of^′t Hooft (ap-

plied originally to the local conformal symmetry) “this

It explicitly contains the elasticity tetrads E^aμ. The

could be a way to make distance and time scales rel-

integer coefficients N_a are three topological invariants

ative, so that what was dubbed as “small distances”

in terms of the Green’s functions:

ceases to have an absolute meaning” [66]. This suggests

∫

∞

∫

that the dimensionless physics can be the natural con-

sequence of the diffeomorphism invariance, and can be

N_a =

ϵ_ijk

dω dS^ia ×

8π²

the general property of any gravity emerging in quan-

-∞ BZ

tum vacuum. The dimensionless physics leads to new

× Tr[(G_ω G-1)(Gki G-1)(Gkj G-1)] .

(14)

topological terms in action with the universal integer

valued topological quantum numbers of the quantum

Invariants N_a describe the quantized response of Hall

vacuum.

conductivity to crystal deformations:

The universality takes place only for the topologi-

cal numbers and the symmetry parameters. The other

dσ_ij

e²

dimensionless quantitites are not universal, being de-

ϵ_ijkN_a .

(15)

dE^ak

2πh

scribed by the functions of the ratios of different mass

scales. In this respect, the answer to the question of

In terms of the conventional tetrads, the gravita-

how many fundamental constants are there in physics

tional Nieh-Yan anomaly related to torsion [51-61],

[67-70] can be trivial: there are no fundamental con-

stants, and the ratios of parameters and the ratio of the

(

)

∂_μjμ5 = λ²

Ta ∧T_a -e^a ∧e^b ∧R_ab

(16)

length scales are the only meaningful quantities [10].

819

16*

G. E. Volovik

ЖЭТФ, том 159, вып. 4, 2021

Funding. This work has been supported by the

20.

F. W. Hehl and Y. N. Obukhov, Ann. de la Fond.

European Research Council (ERC) under the Euro-

Louis de Broglie 32, 157 (2007).

pean Union^′s Horizon 2020 research and innovation

21.

F. R. Klinkhamer and G. E. Volovik, JETP Lett. 109,

programme (Grant Agreement No. 694248).

362 (2019).

22.

M. A. Zubkov, arXiv:1909.08412.

The full text of this paper is published in the English

version of JETP.

23.

A. F. Andreev and I. M. Lifshitz, JETP 29, 1107

(1969).

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I. E. Dzyaloshinskii, P. S. Kondratenko, and

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ЖЭТФ, том 159, вып. 4, 2021

Dimensionless physics

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