The Milgromian acceleration and
the cosmological constant from
precanonical quantum gravity∗
Igor V. Kanatchikov and Valery A. Kholodnyi
Abstract. We show that the Milgromian acceleration of MOND and the
cosmological constant can be understood and quantified as the effects
of quantum fluctuations of spin connection which are described by precanonical quantum gravity put forward by one of us earlier. We also
show that a MOND-like modification of Newtonian dynamics at small
accelerations emerges from this picture in the non-relativistic approximation.
Mathematics Subject Classification (2010). Primary 83C45, 81Sxx, 81T70,
85A99; Secondary 81T99.
Keywords. Cosmological constant, MOND, quantum gravity, precanonical quantization.
Contents
1. Introduction
2. Precanonical quantum tetrad gravity
3. Quantum wave states of Minkowski spacetime
4. The cosmological constant
5. Toward MOND from quantum gravity
6. Numerical estimates
7. Conclusion
Acknowledgment
References
∗ Based
2
3
4
5
5
7
7
8
8
on our presentations at MOND40 (St. Andrews, UK), XL WGMP (Bialowieza,
Poland), EREP 2023 (Bilbao, Spain), DΛrk Energy (Frascati, Italy) and POTOR 9th
(Krakow, Poland) in June-September 2023.
2
Kanatchikov and Kholodnyi
1. Introduction
The precanonical field quantization [1–5], which is based on the De DonderWeyl (DDW) Hamiltonian theory known in the calculus of variations [6], has
been recently applied to general relativity in metric variables [7–10], Palatini
vielbein gravity [11–15], and the teleparallel equivalent of general relativity
[16, 17]. In this formulation, the space-time decomposition and the infinitedimensional canonical Hamiltonian treatment are not a prerequisite for field
quantization. The procedure of precanonical quantization is based on the
Dirac quantization of the Heisenberg-like subalgebra of Poisson-Gerstenhaber
brackets of differential forms that represent dynamical variables within the
DDW Hamiltonian formulation [5, 18–20] and their generalization to the
DDW Hamiltonian systems with constraints [21]. The result is a hypercomplex generalization of quantum mechanics to field theory where quantum
fields are described in terms of the wave functions on the bundle of field variables over space-time which take values in the Clifford algebra of space-time
and satisfy the covariant analogue of the Schrödinger equation whose most
general form reads
⁄
/ − H)Ψ = 0,
(iℏκ
∇
(1)
where Ĥ is the operator of the DDW Hamiltonian function defined from the
Lagrangian function L = L(ϕa , ∂µ ϕa , xν ) as follows:
H := ∂µ ϕa pµϕa − L,
pµϕa :=
∂L
,
∂∂µ ϕa
(2)
/̂ is the covariant Dirac operator on the spacetime (with the Dirac matri∇
ces and/or spin connection becoming differential operators in the context
of quantum gravity [11–17]) and the parameter κ is an ultraviolet quantity
of the dimension of the inverse spatial volume which appears on purely dimensional grounds. The parameter κ also appears in the representations of
operators that correspond to differential forms, e.g. the 3-dimensional volume element dx := dx1 ∧ dx2 ∧ dx3 in 4-dimensional Minkowski spacetime is
mapped to
1
(3)
dx 7→ γ 0 ,
κ
where γ I are the flat spacetime Dirac matrices, γ I γ J +γ J γ I = 2η IJ (I, J =
0, 1, 2, 3). With γ µ := eµI γ I the operators of polymomenta pµϕa defined in (2)
typically have the form
∂
p̂µϕa = −iℏκγ µ a .
(4)
∂ϕ
Considering the connection of precanonical formulation with standard
QFT in the Schrödinger representation [22–27], we conclude that the latter
can be viewed as a specific limiting case of the former, which corresponds
to the infinitesimal value of 1/κ or, more precisely, to the inverse of the
“Chevalley map” (3) (cf.[28]). In this case, the Schrödinger wave functional
Ψ([ϕ(x)], t) can be expressed as a trace of the multidimensional Volterra
Milgromian acceleration and the cosmological constant
3
product integral [29] of precanonical wave functions Ψ(ϕ, x) restricted to the
surface of initial data Σ at the moment of time t, Σ : (ϕ = ϕ(x), x, t). It
was also shown that, in this limiting case, the Schrödinger equation for the
time evolution of the functional Ψ([ϕ(x)], t) = Ψ([Ψ|Σ (ϕ(x), x, t)], [ϕ(x)])
follows from the precanonical Schrödinger equation (1) without resorting to
the standard procedure of canonical quantization.
In this paper, based on a previous work by one of us, we first recall
the results of the precanonical quantization of tetrad general relativity in
Sect. 2, and then, in Sect. 3, discuss the simplest solution of the precanonical
Schrödinger equation for quantum gravity which corresponds to the quantum
analogue of the Minkowski space-time. This allows us to discuss the emergence of the acceleration threshold a∗ in Sect. 3, the cosmological constant
in Sect. 4, and a modification of the Newtonian dynamics, and its relation to
the Milgromian MOND in Sect. 5. The numerical values of the quantities in
question are discussed in Sect. 6 and the conclusions are drawn in Sect. 7.
2. Precanonical quantum tetrad gravity
Let us recall the key results of precanonical quantization of tetrad gravity
based on the previous work by one of us [11–14]. Starting from the standard
Palatini Lagrangian density for general relativity in tetrad variables
L=
ä
ä
1 Ä α βÄ
IJ
IK
eeI eJ ∂[α ωβ]
+ ω[α
ωβ]K J − Λe ,
8πG
(5)
IK
where the tetrad coefficients eα
I and the spin connection coefficients ωα are
I
the independent field variables, e := det(eµ ), and by following the procedure
of the DDW Hamiltonian formulation, we define the polymomenta of the field
variables e and ω, that leads to the primary constraints (in the sense of the
DDW Hamiltonian formulation), viz.,
pα
eI :=
β
∂L
≈ 0,
∂ ∂α eIβ
pα
ω IJ :=
β
∂L
1
[α β]
≈
ee e ,
8πG I J
∂ ∂α ωβIJ
(6)
and the DDW Hamiltonian density on the surface of constraints, viz.,
IK
J
eH := pω ∂ω + pe ∂e − L ≈ −pα
ω IJ ωα ωβK +
β
1
Λe.
8πG
(7)
The analysis of constraints according to [21] identifies them as second-class,
and the calculation of the corresponding generalized Dirac brackets of forms
[21] leads to very simple expressions such as (υα := ∂α dx0 ∧dx1 ∧dx2 ∧dx3 )
′
D
{[pα
e , e υα′ ]} = 0,
′
′
D
α ω
{[pα
ω , ω υβ ]} = δβ δω ,
D
{[pα
e , pω υα′ ]} =
D
{[pα
e , ωυα′ ]} =
′
D
{[pα
ω , e υα′ ]} =
(8)
0.
4
Kanatchikov and Kholodnyi
Ÿ
Using the generalized Dirac’s quantization rule [Â, B̂] = −iℏe{
[A, B ]}D ,
the following representations of relevant operators can be obtained:
∂
(9)
êβI = −8πGℏκiγ J IJ ,
∂ωβ
ĝ µν = −(8πG)2 ℏ2 κ 2 η IK η JL ∂ωµIJ ∂ωνKL ,
(10)
1
∂
∂
+
Λ,
8πG
∂ωβKL ∂ωαIJ
ã
Å
↔
1
∂µ + ωµKL γ KL ∨ ,
4
“ = 8πGℏ2 κ 2 γ IJ ω KM ωβM L
H
α
̸“
∇ = −8πiGℏκγ IJ
∂
∂ωµIJ
(11)
(12)
where the spin connection term in (12) acts on the Clifford-valued precanon↔
ical wave function Ψ(ω, x) by the commutator Clifford product: γ IJ ∨ Ψ :=
1
IJ
2 γ , Ψ . The operators act on Clifford-algebra-valued precanonical wave
functions on the configuration bundle of spin connection variables over spacetime variables, Ψ(ω, x), whose invariant scalar product has the form
Z Y
⟨Φ|Ψ⟩ := Tr
dωµIJ Φ ê−6 Ψ ,
(13)
µ,I,J
µ
µ
◊
where Φ := γ 0 Φ† γ 0 and ê−1 := det(e
I ) is constructed from êI in (9).
Using (11) and (12), the precanonical Schrödinger equation for quantum
gravity, eq. (1), takes the form
↔
∂ 1
∂
K
γ IJ IJ ∂µ + ωµKL γ KL ∨ − KL ωµM
ωβM L Ψ(ω, x)−λΨ(ω, x) = 0, (14)
∂ωµ
4
∂ωβ
where all the physical constants and the parameter κ of precanonical quantization are absorbed in the single dimensionless quantity
Λ
λ :=
.
(15)
(8πGℏκ)2
Thus, precanonical quantization leads to the spin connection foam picture of quantum geometry which is described by Clifford-algebra-valued amplitudes Ψ(ω, x) that obey (14) and the transition amplitudes ⟨ω, x|ω ′ , x′ ⟩
which are Green’s functions of equation (14).
3. Quantum wave states of Minkowski spacetime
The Minkowski spacetime in Cartesian coordinates can be characterized by
ωµIJ = 0 (cf. [31]). In this case, eq. (14) with Λ = 0 reduces to
γ IJ ∂ωαIJ ∂α Ψ = 0,
which is solvable by plane waves Ψ ∼ eikµ x
in the anisotropic dispersion relation
µ
α
IJ
+iπIJ
ωα
α
kα πIJ
kβ π β IJ = 0.
(16)
α
Ψ̃(πIJ
, kµ ) and results
(17)
Milgromian acceleration and the cosmological constant
The required correspondence to the Minkowski spacetime on average
Z
µν
⟨ĝ ⟩(x) = Tr d24 ω Ψ(ω, x)ê−6 ĝ µν Ψ(ω, x) = η µν ,
5
(18)
where the operator ĝ µν is given by (10), is satisfied by the sufficient condition
ĝ µν Ψ(ω, x) = η µν Ψ(ω, x),
(19)
1
η µν ,
(8πGℏκ)2
(20)
which leads to
µ
πIJ
π ν IJ =
and, in turn, using the dispersion relation (17), to kµ k µ = 0. Therefore, the
quantum counterpart of Minkowski spacetime corresponds to the light-like
modes of the precanonical wave function along the spacetime dimensions,
and the massive modes along the dimensions of spin connection coefficients.
The range of those modes in the space of spin connection coefficients defines
an invariant scale of accelerations (in the units in which c = 1)
a∗ := 8πGℏκ.
(21)
At this scale, the classical notion of inertial frames is violated by quantum
fluctuations of spin connection, and the laws of dynamics in external fields
can be modified at small accelerations of the order of or smaller than a∗ .
4. The cosmological constant
From (15),
Λ = λ(8πGℏκ)2 ,
(22)
where the constant λ depends on the ordering of operators in (14). The ordering is fixed by requiring the terms in (14) which do not contain the spacetime
derivatives ∂µ to be symmetric operators on the space of Clifford-valued wave
functions equipped with the scalar product (13) [32]. The contribution from
↔
the Weyl ordered spin connection operator 14 γ IJ ∂ωµIJ ωµKL γ KL ∨ is
î
ó
1 IJ
γ γ KL ∂ωµIJ , ωµKL = 3,
16
compare with the estimation in [31].
λ=−
(23)
5. Toward MOND from quantum gravity
Let us consider the non-relativistic motion of a test particle in a gravitational
field due to a point mass M and taking into account the fluctuating spin
connection of Minkowski spacetime. In the non-relativistic limit,
ẍi = −Γi00 = −ω0i0 = −GM
xi
− ω̃ i ,
r3
⟨ω̃ i ⟩ = 0.
(24)
6
Kanatchikov and Kholodnyi
The fluctuations of spin connections ω̃ i are distributed according to the wave
function Ψ(ω, x) that obeys the non-relativistic approximation of (19):
η ij ∂ω̃i ∂ω̃j Ψ = −
1
Ψ,
(8πGℏκ)2
(25)
whose real-valued ground state (Yukawa) solution and its normalization are
p
1
Ψ= √
e−ω/(8πGℏκ) , ω := ω̃ i ω̃ i ,
(26)
π 8Gℏκ ω
Z
⟨Ψ|Ψ⟩ = dω̃ 1 dω̃ 2 dω̃ 3 ΨΨ = 1.
(27)
By averaging the square of (24) by using ⟨ω̃ i ⟩ = 0 and
Z
1
1
⟨ω̃ i ω̃ i ⟩ := ā2 = dω̃ 1 dω̃ 2 dω̃ 3 Ψω 2 Ψ = (8πGℏκ)2 = a2∗ ,
2
2
(28)
and, by denoting ⟨ẍi ẍi ⟩ =: a2 , we obtain a modification of Newton’s law due
to quantum fluctuations of spin connection:
p
GM
=
a2 + ā2 .
(29)
r2
p
GM /ā where a2 > 0. At larger
This formula is valid in the region r <
distances, the fluctuations of spin connection which lead both to the acceleration threshold (21) and the cosmological constant (22) dominate and hence
violate the approximation of global Minkowski spacetime on average.
When a ≫ ā, we obtain from (29) a corrected Newton’s law
…
ā2
GM
2GM
a+
= 2
for r <
.
(30)
2a
r
3ā
When ā ≫ a, we obtain a MOND-like relation (cf. [33])
…
…
GM
2GM
GM
a2
= 2
for
< r <
ā +
2ā
r
3ā
ā
if the Milgromian acceleration a0 is identified with 2ā. In this case,
»
√
a0 = 2ā = 8 2πGℏκ = 2Λ/λ.
(31)
(32)
Note that our consideration here neglects the influence of the fluctuations of spin connection on the central mass M and quantum correlations
of spin connections at the locations of the mass M and the test particle.
We expect that by taking those effects into account we can obtain the Milgromian MOND together with a realistic interpolating function µ(a/a0 ) and
relate the Milgromian a0 with theoretically predicted scales a∗ and ā more
precisely (cf.[32]).
√ This may lead to a more realistic coefficient in
pthe relation
between a0 and Λ than the one in (32) (cf. Milgrom’s 2πa0 = Λ/3 in [33])
and a more realistic range of MOND-like dynamics than in (31) (cf. Sect. 6).
Milgromian acceleration and the cosmological constant
7
6. Numerical estimates
The numerical values of a∗ and Λ depend on the value of the parameter
κ introduced by precanonical quantization. It is shown in [30] that the
mass gap of quantum Yang-Mills gauge theory is related to the scale of κ:
∆m ∼ (g 2 ℏ4 κ)1/3 . In QCD, ∆m ∼ 100±1 GeV and g 2 ∼ 100 (see [17] and
the references therein), and, therefore, by taking into account all the current
uncertainties in the values of ∆m, the gauge coupling g, and the spectral
estimate in [30], we conclude that κ ∼ 100±6 GeV3 . Consequently, from (21)
and (22), we obtain
a∗ ∼ 10−23±6 cm−1
and
Λ ∼ 10−46±12 cm−2 .
(33)
These values overlap with the values of the Milgromian acceleration a0 ≈
10−29 cm−1 and the cosmological constant Λ ≈ 10−56 cm−2 , respectively.
Correspondingly,
the corrected Newton’s law (30) is valid in the Solar
p
System up to 4GM⊙ /3a0 ∼ 6 × 103 au, i.e. the inner edge of the Öpik-Oort
cloud. For a galaxy of total mass M ∼ 1011 M⊙ , the MOND-like dynamics
in
range of galactocentric distances between
p
p (31) is valid only in a narrow
4GM/3a0 ∼ 11 kpc and 2GM/a0 ∼ 13 kpc. The agreement with the
range of flat rotation curves of galaxies that are described by MOND [34] may
be improved by taking into account the effects listed in the end of Sect. 5.
7. Conclusion
We found two manifestations of quantum fluctuations
of spin connection re√
lated to the Milgromian acceleration a0 = 2a∗ = 2ā: the range a∗ in the
spin connection space of precanonical wave function that corresponds to the
quantum analogue of Minkowski spacetime, and the standard deviation ā of
the distribution of spin connections given by the precanonical wave function
in the non-relativistic approximation. The cosmological constant also appears
as a manifestation of the quantum dynamics of spin connection in the form
of the appropriate ordering of the spin connection operator in precanonical
Schrödinger equation (14). The relation√between the cosmological constant
and the Milgromian acceleration: a0 ∼ Λ appears as an elementary consequence of precanonical quantum gravity. The numerical values of a0 and Λ
can be obtained if the parameter κ has a hadronic scale, which is consistent
with the estimation of the gap in the spectrum of the DDW Hamiltonian
operator of pure gauge theory in [30]. It is also shown that quantum fluctuations of spin connection lead to a modification of Newtonian dynamics
similar to MOND in the regime of very weak gravitational fields. Given the
phenomenological success of MOND [34] and the relation of the Milgromian
acceleration a0 to the cosmological constant Λ and the Hubble constant H
[35], as well as the natural appearance of the Milgromian acceleration and
the cosmological constant in precanonical quantum gravity and its ability to
obtain realistic values of a0 and Λ, albeit with the current error of several
orders of magnitude, we believe that the current trend of introducing ad hoc
8
Kanatchikov and Kholodnyi
entities as various candidates for dark matter, dark energy, and modified theories of gravity in the context of galactic dynamics and cosmology is worth
reconsidering.
Acknowledgment
We thank Ilya Kholodnyi for his help with editing the English of the paper.
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Igor V. Kanatchikov
National Quantum Information Centre KCIK, Gdansk, Poland
IAS-Archimedes Project, Saint-Raphaël, Côte d’Azur, France
e-mail: kanattsi@gmail.com
Valery A. Kholodnyi
WPI, Vienna, Austria & Unyxon, Woodforest, TX, USA
e-mail: valery.kholodnyi@gmail.com