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arXiv:1303.6539v3 [physics.gen-ph] 30 Mar 2016
Quantum Gravity Framework 2.0 : A Complete Dynamical
Framework of Principles for Quantization of General Relativity∗
Suresh Maran
sureshmaran@qstaf.com
March 31, 2016
Abstract
Developing Planck scale physics requires addressing problem of time, quantum reduction, determinism
and continuum limit. In this article on the already known foundations of quantum mechanics, a set of
proposals of dynamics is built on fully constrained discrete models: 1) Self-Evolution - Flow of time in the
phase space in a single point system, 2) Local Measurement by Local Reduction through quantum diffusion
theory, quantum diffusion equation is rederived with different assumptions, 3) Quantum Evolution of a
MultiPoint Discrete Manifold of systems through a foliation chosen dynamically, and 4) Continuum
Limit, and Determinism are enforced by adding terms and averaging to the action. The proposals are
applied to the various physical scenarios such as: 1) Minisuperspace reduced cosmology of isotropic
and homogenous universe with scalar field, 2) Expanding universe with perturbation, and 3) Newtonian
universe. Ways to experimentally test the theory is discussed. This article is a further revision of its
previous revision.
Contents
Contents 1
1 Overview 3
1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Outline of the Paper and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 A Framework of Proposals 4
2.1 Self Evolution in a Single Point Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 The Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Recovering the Usual Classical Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Algorithm for Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Application to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Quantum Reduction for Single Point Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Quantum Evolution and Reduction for Multipoint Universe . . . . . . . . . . . . . . . . . . . 13
2.3.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Algorithm for Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Deterministic, Continuum Limit and Scale invariance . . . . . . . . . . . . . . . . . . . . . . . 18
∗We refer to www.qstaf.com or twitter.com/qstaf for further discussions. The official website for this research is
www.qstaf.com. and the author can be contacted through the website or the twitter page or using email sureshmaran@qstaf.com.
Please signup for our newsletter at www.qstaf.com.
1
3 Applications 21
3.1 Cosmological Reduced Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Cosmology with Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Newtonian Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Experimental Test 26
5 Discussion 28
6 Conclusion 29
7 Acknowledgements 29
Appendices 29
A Derivation Self-Evolution Hamiltonian 29
B An Application of Self-Evolution Equations 32
C A Derivation of Quantum Diffusion Equations 32
D Bidirectional Time Evolution 35
E Transverse-Longitudinal split of metric 35
F Linearized Constraints 37
G WKB Approximation and Momentum Time 38
References 39
2
1 Overview
1.1 The Problem
The purpose of this paper is to address problems related to combining quantum mechanics and general
relativity. They are as follows:
1. Problem of time: Physics is used to make predictions. Knowing the past, physics helps predict the
future. Without time subject of physics is useless. In classical gravity time-evolution clearly is present. The
Wheeler-Dewitt equation, which is the fundamental equation for quantum gravity, contains time evolution
internally, and we need a way to extract time evolution from it in a quantum general relativistic context.
2. Since general relativity is a description of the macroscopic universe, quantum general relativity requires
addressing the quantum reduction problem. So developing a theory of quantum gravity requires including
quantum reduction theory. Usually the difficulty maintaining superposition closely relates to the mass of
the quantum system. Higher the energy of a particle or photon more it acts with particle nature. For
example a proton, which is massive, is a smaller wave-packet than an electron, even though they have equal
charge. This suggests strength of gravitational field may be related to the size of a wave-packet. And so the
gravitational field associated need to be linked to appearance of decoherence. For example, Roger Penrose [5]
have already promoting gravity as linked to quantum reduction. A systematic theory for quantum reduction
using gravity is not yet available and it needs to be developed.
3. Quantum States of extended matter are not covariant objects, and they depend on the space-time
foliation. So quantum reduction depends on foliation. We need a way to objectively describe the quantum
reductive evolution and the issue of foliation on the manifold on which the process depends.
4. Because of non-perturbative nature of quantum gravity one may need to approach it like lattice gauge
theory. This leads to discretizing space-time and fields. Also quantum reduction leads to randomness. These
leads to problems in recovering back the continuum physics. So we need a way to recover back continuum
classical physics addressing the issues of smoothness and determinism.
The goal of this article is to outline a framework of proposals for Planck scale physics, such that it 1)
has time (dynamics), quantum reduction and continuum limit as integral parts of the basic foundations, 2)
is simple and intuitive, 3) has proper physical motivation, 4) is based on simple scientifically established
notions and concepts, and 5) makes minimal assumptions.
The proposals are slightly heuristic, and details may have errors, inconsistencies as of now. But given the
complex task of putting together numerous conceptual and technical elements, certain extent of heuristics and
errors are unavoidable. There may be also conceptual errors. We also need to make simplifying assumptions,
for example in this paper gauge and diffeomorphism constraints are assume to be solved. I have adopted the
versioning system. All these issues will be systematically removed with each new version. The next refined
version is already in preparation.
The framework proposed in this paper is conceptually different, more detailed and advanced from its
previous version [1] in many ways. You can read about the changes in the official website [2]
1.2 Outline of the Paper and Assumptions
In section 2, I elaborately discuss the proposals for dynamics, which are conceptual in nature. The proposals
discussed there apply to the discrete models that are constrained by the Hamiltonian constraint only. This
means, we assume, we have already solved the gauge constraints and diffeomorphism constraints, by restrict-
ing the kinematical Hilbert space. We will address these two issues in the future updates. First a proposal
for quantum evolution of a single point system with quantum variables is discussed. Then I discuss the in-
clusion of quantum reduction to the single point universe through Bloch (Lindblad) equations and quantum
diffusion theory. Then I discuss multipoint universe such as physics on a manifold. Quantum reduction using
quantum diffusion or Bloch equation requires a preferred foliation. So a proposal is introduced to obtain
a preferred foliation through minimizing a functional (to be chosen experimentally) which depend on the
dynamical variables of gravitational fields. Various possible choices for the functional is given. Finally, a
proposal is introduced to enforce smoothness and determinism by adding an extra term and averaging to
the action. Various possible choices for the extra term is given.
In section 3, I discuss how to put together all the four proposals. In section 4, I apply the proposals
to various physical scenarios such as: 1) Minisuperspace reduced cosmology of Isotropic and homogenous
3
universe with scalar field, 2) Expanding universe with perturbation, 3) Newtonian Universe. In section 5,
ways to experimentally test the theory is discussed.
This article aims to develop a framework of proposals, which are more of guidelines, to develop Planck
scale physics with time and reduction. It is precisely not quantum general relativity based on conventional
principles, because some aspects of the conceptual proposals of quantum mechanics and general relativity are
not subsumed, such as unitarity and foliation independence. Primarily this article was intended to address
the issues of time and reduction, by developing a theory of Planck scale process. But since the framework
is built on discrete models, a fourth proposal is added to impose continuum limit. But the framework aims
to reproduce general relativity and quantum mechanics consistent with the experiments at the appropriate
scales. Not all the elements of the framework is new, but some are already well known, such as Bloch and
quantum diffusion equations. But I want to embed them in a proper framework so that they can be used to
develop Planck scale physics.
I assume that the metric in the internal configuration space is positive definite unless specified. We follow
the following conventions in this article: 1) It is assumed that ¯h=c=G= 1, unless specified 2) Einstein’s
summation convention is assumed over indices of internal spaces.
2 A Framework of Proposals
2.1 Self Evolution in a Single Point Universe
2.1.1 The Theory
Consider a single point universe, with one simple quantum system living on it. Assume that simple quantum
system is described by a Hamiltonian constraint only. Let the internal configuration space of the quantum
system is of dimension d, and is made of canonical variables pαand qα. Let mαβ, a function of qα,is the
metric in the internal configuration space. Hereafter I will use mαβ and its inverse mαβ (assuming it exists),
to raise and lower indices. Let me define a scalar product using the metric:
< a, b >=aαbβmαβ .
I will assume mαβ is positive definite for now. The Langrangian as usual is
L(pα, qα, N ) = pα˙qα−NH(pβ, qγ).
Let me assume that a typical Hamiltonian is as follows (without the Lapse):
H(pα, qα) = < p, p >
2+V(qα) = mαβpαpβ+V(qγ) (1)
=pαpα
2+V(qγ).(2)
In this case the classical dynamic equations are
dqα
dt =N{H, qα}=N∂H
∂pα
,(3a)
dpα
dt =N{H, pα}=−N∂H
∂qα
.(3b)
The propagator is
G(qα, q′α; ∆t) = 1
(2π)dZexp(ipα∆qα)δ(H(qα, pα))dpD.
4
We see here that the ∆tterm is absent in the right hand side. So the propagator is a function of the
configurational variables only:
G(qα, q′α) = 1
(2π)dZexp(ipα∆qα)δ(H(qα, pα))dpD.(4)
This form of the propagator is what is to be expected. This is because tis a coordinate variable, it can be
changed by an arbitrary (smooth) rescaling of the lapse (temporal diffeomorphism). So the physical evolution
should not depend on it. Practically we read time by reading observables, for example the position of a clock’s
needles. The q’s are the most basic observables and so the physical time may have to be extracted from
them, for example, assuming that one of them acts as an internal time variable. But, this way of choosing
the time variable is arbitrary and sub jective. We need a more objective way to choose the time variable.
We need to define a set of concepts so that we understand time evolution in a fully constrained system.
Consider the configuration space. We need to use the understanding of classical physics to describe the
flow of time in the Hamiltonian constrained system. Various solutions to the problem of time was reviewed
by Isham [17]. In the schemes described there, our approach will be to identify time after quantization.
Usually in the semi-classical approach using the Wheeler-Dewitt equation, WKB wave-function is assumed
to exist. Then using the Hamilton-Jacobi equation derived from it, the phase of the wave-function is related
to time (for example [18], [19]). Our proposal is closely related this. But assumptions of validity of WKB
approximation is too complex. Here we will make no reference to WKB wavefunctions, and try to develop
time evolution by linking the classical trajectory given by equations (3a) and quantum path integral in
equation (4), only assuming that both of them exist.
In classical analysis doing the constrained system analysis is straight forward. It is a trajectory (qα(τ), pα(τ))
in the phase space given by equations (3a). The configuration space is just enough to formulate the propos-
als. Given ˙qα(0), qα(0),we usually have the curve qα(τ) as a solution satisfying the initial conditions. The
tangent to each point is ˙qα(τ).
Consider a smooth trajectory Qα(τ). Consider a D−1 dimensional one parameter family of flat hyper-
plane Sτin the configuration space going through a point Qα(τ) and the normal be vα(τ) = ˙
Qα(τ). Let
me denote this hyperplane by S(vα(τ), Qα(τ)) or in short S(τ). Let qα= (t, qI
s),where tis the coordinate
along the normal vα
s(τ), and qI
sare D−1 coordinates on the hyperplane orthogonal to it. We can use these
coordinates to study physics on these hyperplanes S(vα(τ), Qα(τ)) in the neighborhood of the region near
the point. Let ψτ(qI
s) be a wavefunction on this hyperplane. Then we can derive a propagator for evolving
this wavefunction with respect to the one parameter family of hyperplane as I will discuss next.
If qα∈S(τ),from the equation for the hyperplane we have,
vα(τ)(qα−Qα(τ)) = 0.
Let eα
ibe the global unit vectors, where ivaries from 0 to D−1, with eα
0vα(τ)>0.Let ei
αbe the
inverse of eα
i.Let Eα
Iare the unit vectors on the surface S=S(τ) where Ivaries from 1 to D−1. The
metric on the hyperplane is σαβ = (mαβ −¯vβ¯vα).Consider the figure (1). We can get Eα
Iby projecting eβ
I
to S, Eα
I=σα
βeβ
I. Let Qα(τ) be the origin on S, then the hyperplane coordinate of point in the eβ
Ibasis
is qα
s=Qα(τ)−qα,implying qα
svα= 0.In the Eα
Ibasis the hyperplane coordinates are qI
s=EI
αqα
s,where
EI
β=σα
βeI
α.
Let us associate a normalized wavefunction ψ(qI
s, τ ) to the surface S(τ). If q′α∈S′=S(τ+dτ ),the
propagator between Sto S′is given by,
˜
G(qα, q′α;S, S′) = 1
(2π)dZvs,αpα<0
exp(ipα(qα−q′α))δ(H)dpD.(5a)
I have used the condition vs,α pα<0 to define a casual propagator, such that the time evolution along vα.
5
e
i
q( )
s
q( )
E( )
I
S( )
surface
( )(q-Q( ))=0
()
Figure 1: Surface S(τ) and various basis vectors.
Let me assume the width of the wavefunction ∆qs(τ),defined by
< qI
s(τ)>=ZqI
s|ψ(qJ
s, τ )|2dqD−1
s,(6a)
∆qs(τ)2=Z(qI
s(τ)−< qI
s(τ)>)(qs,I (τ)−< qs,I (τ)>)|ψ(qJ
s, τ )|2dqD−1
s,(6b)
remains much smaller than the radius of curvature of Q(τ),illustrated in figure (??). Then qI
sare good
coordinates to study the evolution of the ψ(qI
s, τ ).
˜
G(S, S′) be the operator form of ˜
G(qI, q′I;S, S′) in the qI, q′Icoordinates. The propagator needs to be
normalized as follows:
G(S, S′) = lim
S′′ −>S
˜
G(S, S′)˜
G(S, S′′ )−1.(7)
This renormalization removes factors of integration from δ(H), so that limS′−>S G(qI, q′I;S, S′) = δ(qI−q′I).
Proposal 1: Self-Evolution: The temporal flow of time in a quantum Hamiltonian constrained system is
described by 1) Wavefunction ψτ(qI
s)on the one parameter family of hyperplanes S(τ) = Sτ(˙
Qα(τ), Qα(τ)),
for a given Qα(τ),2) The propagator between the wavefunctions on the hyperplane given by equation (7), 3)
The path Qα(τ)considered as a C1smooth function of τ . 4) Given an arbitrary path Qα(τ),one can evolve
the wavefunction ψτ(qI
s)normal to the hyperplane S(˙
Qα(τ), Qα(τ)).at each instant τ .The physical value of
Qα(τ)is such that < ψτ|ˆpI
sˆpsI + ˆqI
sˆqsI |ψτ>is a minimum for all possible of qα
s(τ).
The S(τ) = Sτ(˙
Qα(τ), Qα(τ)) represents the classical information contained, which can be derived from
the Euler-Langrange Equations or Hamilton equations, while ψτ(qI
s) represents the quantum information.
Then αsis very large, the Qα(τ) evolve close to its classical expected value given by Hamilton equations.
Because of this Ω becomes close to zero. Now this is just a generalization of what a person experiences in
relativistic quantum mechanics. He observes the physics around himself as it happens on the hyperplane
orthonormal to the direction along which he travels, with he being the center.
Given an arbitrary path Qα(τ),one can evolve the wavefunction along ψτ(qI
s) along the hyperplane
Sτ(˙
Qα(τ), Qα(τ)).The physical value of Qα(τ) is such that Ω is a minimum for all possible Qα(τ).But there
6
center of curvature
Q( )
S( )
surface
( )(q-Q( ))=0
q( )
s
width of
R( )
radius of curvature
Figure 2: Evolution of the surface S(τ).
could be more than one solution for this. Also if ∆qs(τ) is two high compared to norm of < qs(τ)>(equation
(6a)) then it is not sensible to think Qα(τ) as the unique classical information associated with the quantum
evolution. Proposal 2 that we will discuss in the next section will improve this situation.
Using this proposal 1 as a guideline, let me derive the Hamiltonian associated to the continuous evolution
of the wavefunction on a parameter family of hyperplanes Sτ. Let me calculate the Hamiltonian for the self-
evolution, from surface S=S(τ) to S′=S(τ+dτ ). The details of the calculation is given in appendix A.
The propagator for the evolution of the wave function from S(τ) to S′=S(τ+dτ) is,
G(q′I
s′, qI
s,∆τ) = Zpα¯vα<0
exp(i(ps
IδqI
s−Hs(ps
I, q′I
s′, Qα(τ))∆τ))dqI
s,(8)
where
Hs(ps
I, qI
s, Qα(τ)) = ps
IqJ
shI
J−pv(|v|+qI
s
d¯
Eα
I
dτ ¯vα),(9)
|v|=|dQα
dτ |,
and
hI
J=¯
EI
α
d¯
Eα
J
dτ .
Assuming mαβ =δαβ. Using
7
pv=q−pI
sps
I−2V(Qα(τ) + qI
s¯
Eα
I),
we can calculate the effective classical equation of motion for the Hamiltonian Hs
˙qI
s=qJ
shI
J−∂pv
∂ps
I
(|v|+qK
s
d¯
Eα
K
dτ ¯vα),(10)
˙ps
J=−ps
IhI
J+∂pv
∂qJ
s
(|v|+qK
s
d¯
Eα
K
dτ ¯vα) + pv(d¯
Eα
J
dτ ¯vα).(11)
Examples of application of these equations is given in Appendix A.
2.1.2 Recovering the Usual Classical Motion
Let me show that we can recover the classical Hamilton motion from the equations (10) and (11) with the
condition of minimality of Ω. Let σαβ =δαβ. Let initially qI
s= 0 and ps
I= 0.
pv=q−pI
sps
I−2V(Qα(τ) + qI
s¯
Eα
I),
∂pv
∂ps
I
=−pI
s
pv
,
∂pv
∂qJ
s
=−1
pv
∂V (Qα(τ) + qI
s¯
Eα
I)
∂qJ
s
.
Using the initial conditions qI
s= 0 and ps
I= 0, we need to solve for ˙qI
s= 0 and ˙ps
J= 0 to get the equations
of motion for Qα(τ).Using these,
pv=p−2V(Qα(τ)),
˙qI
s= 0 =⇒ − ∂pv
∂pI
(|v|) = 0,
=⇒pI
s= 0.
˙ps
J= 0 implies,
∂pv
∂qJ
s
(|v|) + pv(d¯
Eα
J
dτ ¯vα) = 0,
−1
pv
∂V (Qα(τ) + qI
s¯
Eα
I)
∂qJ
s
(|v|) + pv(d¯
Eα
J
dτ ¯vα) = 0,
−1
pv
∂V (Qα(τ) + qI
s¯
Eα
I)
∂qJ
s
(|v|2)−pv(¯
Eα
J
dvα
dτ ) = 0.
Using initially ¯
Eα
J=δα
Jfor α6= 0,
−|v|2
(pv)2
∂V (Qα)
∂Qα−(d2Qα
dτ2) = 0.
8
Since τis a arbitrary parameter vis defined upto an arbitrary scale. So by setting |v|=pv,we get,
d2Qα
dτ2=−∂V (Qα)
∂Qα, α 6= 0.
Evolution of Q0is determined by the condition |v|=pv, which is equivalent to the Hamiltonian constraint.
This and the above equation both are equivalent to equations (3a).
2.1.3 Algorithm for Evolution
Let me write out the algorithm for evolving the wavefunction using proposal 1:
1. First a wavefunction ψ(qI
s, τ ) is given on initial hyperplane S(τ) with normal vector να(τ).Let the
expectation value of the wavefunction be Qα(τ) in the global coordinates. Let Qα(τ) be the origin of
the coordinates qI
sin S(τ). Let < ps
I>be pαin global coordinates. Let me assume that the norm of
pαis small compared that of να(τ).
2. Evolve the wavefunction ψ(qI
s, τ ) along να(τ) to a new hyperplane S(τ+dτ) with normal να(τ+dτ ) =
mαβpαgoing through point Qα(τ)+dτ να(τ). Set Qα(τ+dτ ) = Qα(τ)+dτ ν α(τ) as origin of coordinates
on the new hyperplane. The Propagator and Hamiltonian for evolution was given in equation (8) and
equation (9). Using equation (9), the modified Schr¨odinger equation is
dψ(qI
s, τ ) = iHs(ps
I, qI
s, Qα(τ))dτ . (12)
3. Repeat steps 1 and 2.
The above evolution ensures minimization of < ψτ|ˆpI
sˆpsI + ˆqI
sˆqsI |ψτ>at each step.
2.1.4 Application to Physics
Let me apply discuss the application of the theory to general relativity coupled to electromagnetic field
and scalar field. The canonical coordinates are (hab , πab ) for gravity, (Aa, Ea) of Electromagnetic field, and
(φ, χ) for the scalar field. Since we are studying a single point system, without the interaction terms, the
Hamiltonian constraint is
Hnon−int = + 1
2χ2+m2hφ2+1
2(E2)−1
cg
(π2−πabπab ),
where cgis the gravitational coupling constant. Now our basis is made of tensors and scalars. Our basis
eα
iis actually made of collection different tensor bases: dxa⊗dxb,dxaand 1,where 1 is the basis for
scalar field. They belong to different spaces. But it does physically make sense in unified theories such as
in Kaluza-Klein theory, where tensor, vector and scalar fields becomes components of higher dimensional
tensor, and in string theory the fields are just various string components. Nevertheless to apply the method
discussed in this section, we need to name the 10 pairs of conjugate variables, as qαand pα.For example:
q0=φ, q2=A1, q3=A2, q4=A3,
q5=h11, q 6=h22, q7=h33, q8=h12, q 9=h23, q10 =h32.
p0=φ, p2=E1, p3=E2, p4=E3,
p5=π11, p6=π22 , p7=π33, p8=π12, p9=π23 , p10 =π32 .
9
Some of these terms are redundant because the diffeomorphism constraint, and gauge constraints need
to be imposed, which is another separate problem. In section three and four, this case will be discussed, but
essentially we will postpone it to future versions or separate paper.
The Hamiltonian constraint can be rewritten in terms of the pand qvariables and the theory we developed
in this section can be applied. The kinetic term in term of p′sand q′swould be too complicated for display,
unless we use some unified theory of fields. So I don’t explicitly show it here. Computer simulation to analyze
the behavior of such a system, where all the relevant variables can be directly entered into the program.
2.2 Quantum Reduction for Single Point Systems
(This section is based on Lindblad equation and quantum diffusion equations. Advanced readers can skim
through this section.)
Consider the single point system discussed in the last subsection. The modified Schr¨odinger equation (12)
derived describes the evolution of the system in the direction vα. The equation results in the system evolving
into a macroscopic superposition state. To prevent this we need continuous reduction of the system which
removes the macroscopic superposition. The general form of continuous reduction of a quantum system
is given by the Bloch equations in the Lindblad form [23] governing evolution of density matrix (reviewed
in [24]):
˙ρ=i[ˆρ, ˆ
H] + X
m
(2 ˆ
Lmˆρˆ
L+
m−ˆ
L+
mˆ
Lmˆρ−ˆρˆ
L+
mˆ
Lm),(13)
where ρis the density matrix and Lmare the operators representing observables to be continuously measured.
This equation has been extensively studied and has been useful in various experimental situations [25]. It
describes an ensemble of identical quantum systems and does not tell how each individual system evolves.
It is not the most natural and explicit form to use to describe an individual quantum system. So, we need
to consider the equivalent equation, given by Percival, Gisin and Diosi [8], which describes the stochastic
motion of the quantum system state |ψ > of a quantum system:
d|ψ > =−iˆ
Hdτ |ψ > +X
m
(ˆ
Lm−<ˆ
Lm>)|ψ > dzm√dτ (14)
+X
m
(2 <ˆ
Lm>ˆ
Lm−ˆ
L+
mˆ
Lm−<ˆ
L+
m>< ˆ
Lm>)|ψ > dτ ,
where dτ is the time interval of evolution in the non-relativistic quantum mechanics. The dzmare complex
numbers representing Gaussian distributed independent random variables. More explicitly, the real and
imaginary parts of dzmare Gaussian random variables such that the statistical expectation values are given
by,
M(dzm) = 0, M (dzmdzn) = 0, M (dzmdz∗n) = 2δmn ,(15)
where Mrefers to the statistical mean.
Let me clarify how the third term works a little bit. Consider that |ψ > is expanded as a superposition
of the eigenstates of ˆ
Lm.As |ψ > evolves, the third term tends to reduce the amplitude of an eigenstate in
the sum to the extent to which its eigenvalue is far away from the expectation value of <ˆ
Lm>. Because
of this |ψ > evolve such that the amplitudes of the components are peaked close to <ˆ
Lm>, a semiclassical
state. In equation (14) the second terms randomizes the system, third term classicalizes the system. These
are natural components of macroscopic quantum reduction.
Percival applies this to quantum field theory and indicates that the resultant theory is non-unitary [9].
But, in case of quantum gravity the universe cannot be described by unitary evolution alone because that
would lead to superposition of macroscopic states. Clearly, experimentally, whenever the quantum state of
a system evolves into a superposition of macroscopic quantum states it probabilistically evolves to one of
10
the macroscopic states. So, for a macroscopic universe, the quantum evolution must be described by an
equation that has three components: a deterministic unitary component, a stochastic component, and a
component that prevents macroscopic superposition. The modified Schr¨odinger equation (14) is the most
natural form of it and the three terms in the right hand side of the equation give the necessary components
in the respective order.
In general the stochastic evolution can defined upto a norm of |ψτ>. The norm of |ψτ>is not physically
relevant. The physical interpretation of the theory comes through its relation to density matrix which doesn’t
depend on the norm of |ψτ>
ρ=M(|ψτ>< ψτ|)
< ψτ|ψτ>,
which evolves by the Lindblad equation. The Mis the statistical mean with respect to the random variables
z. The eigenstates and eigenvalues of ρgives the possible physical states |ψτ>of the quantum system and
probabilities for observing them respectively.
In the appendix C a general stochastic evolution equation motivated by equation (14) is derived. The
derivation is based on [10], but different in details. Let me briefly summarize the derivation in the appendix
C. A general stochastic evolution equation is
|dψ >=α|ψ > dt +βm|ψ > zm√dt,
where αand βare operators on |ψ >, and zmobeys equation (15). Summation over repeated indices is
assumed.
Let me solve d(< ψ|ψ >) = 0,assuming that dt and zmare free variables, and, αand βare independent
of dt and zm. As shown in the appendix C, the constraints are too strong that they may eliminate the
quantum diffusion equation (14) itself.
We need to take a different route. First let me solve Md(< ψ|ψ >) = 0,assuming αand βare independent
of dt and zm.The general solution for αis
α=iH +γ−< γ > −β+
mβm,(16)
assuming αand βare independent of dt and zm. Here His a Hermitian operator, and γis an arbitrary
operator.
Now let me for solve d(< ψ|ψ >) = 0 assuming, dt are the free variables, α, β are independent of dt.
Further a solution for αcan be obtained by adding a real number to equation (16) to keep the norm constant.
To summarize we have the final form of the dynamics equations are
|dψτ>=α|ψ > dt +βm|ψ > zm√dt, (17)
where
α=iH −γ−β+
mβm+c, (18a)
βm=Lm−< Lm>, (18b)
c=−1
2< β+
mβn>(zm¯zn−2δmn)+ < γ > −<2α+βm> zm√dt. (18c)
The cis a c−number and is a random function of z. The cis also dependent on √dt. I will assume that
zm√dt can be neglected hereafter, unless specified otherwise.
We can derive the evolution equation for ρ=M(|ˆ
ψ >< ˆ
ψ|),from equation (18a) (without neglecting
√dt term). The evolution equation of ρis
11
dρ
dt =ρ(˜α+) + ( ˜α)ρ+βmρβ+
m,(19)
˜α=iH +γ−< γ > −β+
mβm.
Here the √dt term does not show up.
To get the quantum diffusion equation and the Lindblad equation we need to set [10],
βm=Lm−< Lm>,
γ=< Lm> L+
m−Lm< L+
m> .
In this case, using the evolution equation (19) it easy see that the system in long term approaches the
expected probability distribution for Copenhagen interpretation.
Let me state the second proposal of dynamics.
Proposal 2: Local Quantum Reduction -Given a path Qα(τ), the quantum state of a single point
quantum system along with the self-evolution also undergoes continuous reduction with respect to observables
Lithrough semiclassicalization and randomization given by equation below.
d|ψτ>=iHs|ψτ> dτ +X
m
βm|ψτ> dzmq|˙
Qα|∆τ+ (γ−X
m
β+
mβm)|ψτ>|˙
Qα|∆τ, (20)
γ=−σ+< Lm> L+
m−Lm< L+
m> .
with Hsis from equation (9), βi=Li−< Li>, with σis Hermitian operator to subject the evolution to
further reduction.
Physical relevance of σwill be evident in the fourth proposal. Since the cterm is ignored, |ψτ>is not
assumed to be normalized. So
< Li>=< ψτ|Li|ψτ>
< ψτ|ψτ>.
The iHs|ψτ>term is self-time Hamiltonian derived from proposal 1 in equation (12). The ∆τis the time
measure from the first proposal of dynamics, and the operators ˆ
Lmare simple functions of the conjugate
variables ps
Iand qI
sto undergo continuous reduction.
Usually for the applications of the Bloch equation (13) to study the evolution of the density matrix
of a quantum system, the Lm’s are to be determined by what are to be measured in the experimental
context. But, here in the second proposal of dynamics we assume that the Lm’s are fundamental quantities
in quantum gravity to be determined experimentally. The natural and simplest choice for the Lm’s are
given by ps
Iand qI
s, or some simple functions of them. These observables need to be gauge invariant and
diffeomorphism invariant as expected by the theory studied. Introducing diffeomorphism invariance requires
studying multipoint system, which will be discussed in the next to subsections.
The fermionic fields have zero expectation values. So these cannot contribute to the Lm’s. The fermionic
particles can be measured by measuring the bosonic fields they generate. For example, superposition of a
particle wavefunction at different points, results in superposition of fields generated by it such as gravitational
and electromagnetic fields. Continuous reduction of these fields with Lm’s, reduces fermionic fields.
The combined quantum system forms a complete reality by itself and there is no outside observer to
make reduction. The system needs to be understood as undergoing continuous reduction by itself instead of
being considered as undergoing measurement.
There is no necessity that one needs the quantum diffusion theory. It might be simple to just use the
Bloch equations (13). A detailed study of the model might help whether one can just restrict to the density
matrix formalism of the theory. Also since the reference frame with respect to which the quantum diffusion
occurs keeps changing according to proposal one. This might interfere with the reduction process and make
12
quantum diffusion theory problematic for reduction process. But if the reduction occurs faster than the
curving of Qα(τ),which determines the reference frame, then the theory will remain sensible.
The new modified Hamiltonian can be directly included in the algorithm discussed in the last section.
This can randomize the evolution of Qα(τ) and quantum state. This randomness can reduced and smooth
evolution can be reproduced by introducing many-body interaction and continuity conditions in the next
two proposals.
2.3 Quantum Evolution and Reduction for Multipoint Universe
The first proposal of dynamics focused on a quantum system at a single point that evolved according
to a single time parameter. In quantum gravity we want to evolve the quantum states from one spatial
hypersurface to another spatial hypersurface. In a spatial hypersurface there is infinite number of points,
with a quantum system at each point. So let me discuss how to understand time evolution in a many-point
quantum system. If there are many interacting fully constrained quantum systems, then for each point x,
there are one set of conjugate variables px,α,qα
x(Ddimensional space internal space). To each point we
can apply theory discussed in proposal one, then there will one classical curve Qα
x(τx) for each point, one
hyperplane Sx
τ(˙
Qα
x(τx), Qα
x(τx), and one free (dummy) parameter τxfor each point.
Let me assume that space is discretized and is made of countable number of points. Let Bbe the number
of points, and for simplicity let us assume Bis finite. Assume that the quantum system at each point x
is described by an identical Hamiltonian constraint Hxonly, and it has an interaction term that involves
adjacent quantum systems. Each step of the evolution depends on how Qα(τx) varies with ∆τx. Now
consider the propagator defined by proposal 1 in equation (7). For each system at x, we have one curve
Qα
x(τ) assigned. Then the combined one step propagator is
˜
G({qα
s,x, q′β
s′,x;Sx, S ′
x,∀x}; ) = 1
(2π)BD Zvα
s,xpα,x <0,∀xY
x{exp(ipα,x(qα
x−q′α
x))δ(Hx)dpD
x},(21)
G({Sx, S′
x,∀x}) = lim
S′′
x−>Sx,∀x
˜
G({Sx, S′
x,∀x})
˜
G({Sx, S′′
x,∀x}).
The repeated application of the one-step propagator for infinitesimal ∆τxsmoothly evolves all the sys-
tems. The sequence of the quantum states, defines the states of the system at various consecutive instants.
As the combined system evolves the classical expectation value of the momentum and the configuration
variables pα,x and qα
xalso evolve. Let me apply it for Fsteps. From equation (21) we have:
G({qα
x,0, qβ
x,F ,∀x}) = ZF
Y
k=1
[G({qα
x,k, qβ
x,k−1;Sx(τk), Sx(τk−1),∀x})] (22)
(Y
x
F−1
Y
k=I+1
dqx,k).
Here Qα
x(τx) is a free variable at each point of the discretized spacial surface. Applying the principal 1
extremal proposal yields a curve Qα
x(τx) for each point. But since τxis a dummy variable we can fix the
arbitrariness. This done by using the condition |˙
Qα
x(τx)|=pv,x, where pv,x is the classical momentum along
˙
Qα
x(τx). Then ∆τxphysically represents the proper time.
Now let me apply this to physics with gravity, scalar field and vector field. The Hamiltonian constraint
with interaction term is
HT= + 1
2hhab∂aφ∂bφ+χ2+m2hφ2+1
2(E2+B2)−cghR +1
cg
(1
2π2−πabπab ).
As discussed in section 2.1.2 variables can be rewritten in terms of pαand qα. The ultimate expression
13
for the above Hamiltonian constraint will be cumbersome. Important thing to note here is that there will be
one time parameter τxfor each point. Assume that above Hamiltonian constraint is discretized in a cubic
lattice made of Bcubes. Then we can apply the above theory.
Each step of the evolution depends on the values of ∆τx. Let τbe a continuous time parameter, which
varies from τ= 0 to τ=T. Let me define ∆τx=nx(τ) ∆τ, where the nx(τ) are continuous functions of
τ, one of them for each point x. The repeated application of the one-step propagator for infinitesimal ∆τ
evolves the quantum state. The nx(τ) functions defines the various ways to foliate the discrete geometry,
whose topology is Bpoint ⊗1D, described by the above Hamiltonian constraints. nx(τ) is essentially is the
lapse.
Now if we want to include the reduction at each point discussed in proposal 2, it depends on the foliation,
as it is not covariant. Given a foliation described by certain choice of nx(τ),we generalize proposal one and
two, given a path qα
s,x(τ) for each point, the combined quantum state of all points of the manifold can made
to undergo continuous reduction with respect to fundamental field variables Li,x at each point, through
semiclassicalization and randomization given by equation below:
d|ψτ>=iHs,x|ψτ> nx(τ)dτ +X
m,x
βm,x|ψτ> dzm,x q|˙
Qα
s|nx(τ)∆τ+(γx−X
m,x
β+
m,xβm,x )|ψτ> nx(τ)|˙
Qα
s|∆τ.
(23)
where Hs,x is from equation (9), γ x=−σx+< Lm,x > L+
m,x −Lm,x < L+
m,x >, and βi,x =Li,x−< Li,x >,
the suffix xindicates the point to which the quantities corresponds.
In case of general relativity and quantum fields we will have diffeomorphism and gauge constraints to take
care of. If we discretize them then they will become discrete constraints. One can use the above equations
along with these discrete constraints to evolve the multipoint quantum state. Discretization of equations
of general relativity and quantum fields, and using the discrete equations to evolve the system is possible:
for example lattice gauge theory, black hole collisions, casual dynamical triangulations, etc. Inclusion of the
constraints in general case will discussed in future versions of the paper, or in a separate paper.
Now we need to pick a foliation that is be relevant to do the reduction process. Now there are three
questions to be addressed: 1) whether the reduction process occurs along a preferred foliation, 2) what is the
choice of the foliation along which the reduction occurs, and 3) whether this can be addressed as experimental
questions. If reduction happens along a preferred foliation, this process will correlate information along the
hypersurfaces of this foliation. This correlation is a physically measurable effect.
There is no experimental proof that precludes reduction along a preferred foliation. Using preferred
foliation seems to be the simplest choice as of now. So let’s assume the reduction in a preferred foliation in
this paper, and work out the experimental consequences in this and in future papers. Choosing preferred
foliation does not mean introducing a new aether, as long as the foliation depends on gravitational field
itself, as we will see next.
If the answer to the first question is yes, we can try to guess what could be the most natural foliation
along which the reduction might occur. For this, consider the set up used for studying continuum canonical
general relativity. Consider a space time with metric gαβ and one parameter spacial foliation St,where St
is the spacial hypersurface for a given t. This can foliation can be specified by function t(x), x is a point on
space time, with t= constant describes the surface St. We can choose tto be the time coordinate. Consider
the vector field, Tγ= ( ∂
∂t)γ.Tγgenerates a one parameter family of space-time diffeomorphism, such that
a given initial surface St1is mapped to a different surface St2of the foliation. So specifying Tγ, assuming it
is integrable, is another way to define the foliation. The universe is described by combination of ideal space
times such as 1) Homogenous and Isotropic: Robertsen-Walker metric, 2)Static: Schwarzschild metric, 3)
Stationary type: Kerr-Metric, and Reisnesser-Nordstrom metric, listed below:
ds2=a(t)2(dt2−dx2+dy2+dz2),(24)
ds2=N(r)dt2−R(r)dr2−r2dΩ,(25)
14
ds2=N(r, θ, φ)dt2−h(r, θ, φ)abdxadxb−2Nadxadt. (26)
Space-Time Type->Homogenous and Isotropic Static type Stationary type
Metric equation(24) equation(25) equation(26)
Most Natural Foliation t= constant t= constant t= constant
∂
∂t conformal killing conformal killing conformal killing
d
dt ¯
hab 0 0 0
R0 0 6= 0
¯πab 0 0 6= 0
The most natural foliation for each case is given in the table above.
Consider the static case - Schwarzschild space time. The physical information is contained in the distri-
bution of matter and the gravitational field around it. All this information is transferred unchanged along
the foliation in which it is static. Our quantum measurement experiments are usually done along the time
parameter along the time-like killing vector ( ∂
∂t )γ. Any motion of the measurement instruments or earth
itself is too non-relativistic to alter the direction of flow compared to ( ∂
∂t )γ. Also consider the linear gravity.
To first order the gravity is described by gravitational potential. On the Schwarzschild case the gradients
are parallel to the hypersurfaces of the static foliation t=constant. To first order these static surfaces are
the directions along which gravitational forces act. So these hypersurfaces are unique in this way.
In case of the Robertson Walker metric, similar to Schwarzschild metric, most calculations quantum or
semiclassical is done along the foliation given by scale factor as cosmological time. Long distance correlation
in Cosmic Microwave background (CMB) has been derived using quantum correlations during inflation using
scale factor as time. Until now all the observations of Cosmic microwave background is consistent with such
a theoretical analysis.
From the above two case we can seen that the foliation defined by the conformal killing vector ( ∂
∂t )γ
appears to a good candidate. Let me define tensor Cαβ defined as a function of space-time metric gµν by
Cαβ (gµν , T η) = £T(gαβ )−1
4(gγδ £T(gγδ))gαβ ,
where £Tis the lie derivative along Tα.For a vector Tαto be conformal killing, Cαβ is to be zero.
One can see from the table that there are many possible ways to identify the natural foliations. Some of
these are listed below:
1. Trace free momentum ¯πab is zero,
2. The scalar curvature of the hypersurfaces Ris zero,
3. Trace free transverse component of hab or πab is zero,
4. Hypersurface volume V=R<√h > dx3is maximum (not mentioned in the table),
5. If ¯
hab =h−1
3hab, with h= det(hab ),then d¯
hab
dt = 0,and
6. Cαβ (gµν , T η) is zero.
The first four are clearly true for spherical static case and cosmological case. The last two are also
true for these two space times and also for stationary types such as rotating and/or charged case. The real
physical space-time, is a combination of many types of metric and the six conditions hold only approximately.
So we need to consider a physical choice of foliation such that it fits very closely to the natural time-like
hypersurfaces associated to them. Instead of considering the tensors to be zero, we need to specify a norm
like functional on these tensors, to measure how small they are. Let me consider them one by one:
1. Υ(πab) = R¯πab πab
√hdx3.
15
2. Υ(hab) = RR√hdx3.
3. Υ(πab) = RπT T
ab πab
T T
√hdx3, T T stands for trace free transverse component.
4. Υ(hab) = Rh2
3d¯
hab
dt
d¯
hab
dt √hdx3.
5. For measuring the smallness of Cαβ ,consider the most obvious norm:
ZCαβ Cγδ √gd4x=Zgαγ gβδ Cαβ (gµν , T η)Cγδ (gµν , T η)√gd4x
The second line makes the depends on gαγ and Tηto be explicit. Since the metric is Lorentzian, the
measure is not positive definite. So the smallness of RCαβ Cγδ√gd4xdoes not imply the smallness
of components of Cαβ .To surmount this, metric can be Euclideanized so that the norm is positive
definite.
Υ(gµν , T η) = Zgαγ
Egβδ
ECαβ (gE
µν , T η)Cγδ (gE
µν , T η)√gEd4x
where gE
µν is the Euclidean version of the Lorentzian metric gµν , and gµν
Eis the inverse of gE
µν .
6. Υ(hab) = −R<√h > dx3,smallness of this norm measures the largeness of the volume.
7. Also one can consider using the full canonical momentum of gravitational field to define Υ :
Υ = Z(πabπab
cgh)√hdx3
=Z(cg
KabKab
h)√hdx3.
8. One also need to investigate a more general way to define Υ using the kinetic terms of the integral spin
fields, without including Dirac fields (as the field expectation value are zero):
Υ = Z(π2
φ+1
2E2+¯πab ¯πab
cgh)1
√hdx3
where Kαβ is the full extrinsic curvature.But, in most cases, the gravitational term dominates because,
cgis very large.
Now let me propose the following
Proposal 3: Global Quantum Reduction -The quantum evolution and reduction process occurs along
a spatial foliation such that the C1smooth functions nx(τ)take smooth values, such that relative probability
weight is given by exp(−crΥ), where cris a fundamental constant, where Υis one of the measures in the
above list, to be discovered and verified experimentally.
One can assume Υ being minimum is sufficient to determine the foliation. But minimality of Υ may not
necessarily give a unique foliation. That is why I have chosen a statistical form for proposal three. There
are various possible candidates for describing the foliation of the three types of geometries: The first five
choices fits with the canonical form of dynamics. The last choice is covariant. Proposal 3 is for continuum
version of general relativity. We need to develop a discrete version of this proposal to be compatible with
the other three proposals in this paper.
We will discuss this further in upcoming updates and try to derive the entropy of the gravitational field.
When there are mixture of macroscopic bodies evolving in expanding universe, the minimal foliation is
made by merging of various types of foliations. This is illustrated in the figure (3) below. Close to the
cosmic celestial bodies such as planets or stars, the foliation is determined by purely Schwarzschild metric,
16
the minimal surfaces are normal to time-like vector along which the body moves. Between the celestial bodies
these surfaces deform slowly through intermediate foliation, whose normals are some weighted average of the
velocity vectors of the planets depending the position and the masses of the bodies. In between galaxies we
have surfaces described by constant scale factor.
Expanding Universe
Collapsing Matter
Collapsing Matter
time
surfaces of quantum instants
Figure 3: Proposed preferred minimal foliation describing time instants. This is proposed to depend
dynamically on the flow of gravitational field itself. The surfaces are assumed to be microscopically random
spatial hypersurfaces.
Particles such as atoms or elementary particles do not disturb appreciably the gravitational field deter-
mined by large celestial bodies. So they evolve and decohere along these special foliations as I have discussed
in proposal 3. One might think that proposal 3 is not in the spirit of General relativity as it depends on the
foliation. But this not true, because we have not chosen the foliation kinematically. The foliation is chosen
here dynamically based on values of gravitational canonical field variables.
The various alternative proposals for Υ need to be investigated theoretical and experimentally to look
for a precise theory. The continuous reduction of the quantum state correlates the classical and quantum
information along the hypersurfaces. Now for the Υ′ssuggested the correlation happens along the slightly
random hypersurfaces that are close to the physically intuitive ones. This may be a physically observable
effect, as mentioned before, that may have useful consequences. So this effect needs to be studied more
theoretically and/or experimentally. Later we will suggest an experimental test based on this.
2.3.1 Interpretation
The τis defined as a global evolution parameter. The evolution along τcould physically mean three different
things:
1. 3D Evolving block model: This is the Newtonian way of interpretation: τis a global time parameter
along which a curved 3D semi-classical universe undergoes quantum evolution. The past is semiclas-
sical. Future is non existent. The present evolves along a unique foliation of 4D metric, which is
probabilistically chosen from many possible foliations from proposal 3. Consciousness of all observers
evolve along τ.
2. 4D Block Universe Model: τis simply a global foliation parameter of a 4D block universe model.
17
Observers physical time is a proper time parameter along his world line as usually defined in relativity.
But, this interpretation does not explain why observer time flows unlike the first interpretation.
3. Along the foliation defined by proposal three there are some possible unknown physical processes hap-
pen, and moves in time, which determines the quantum reduction and semiclassicalization. Discovering
these physical processes is a future course of research.
If of this may be the right physical interpretation can only be decided by experimental study, if possible.
2.3.2 Algorithm for Evolution
Let me discuss a typical algorithm for evolving a piece of a Planck scale sized spatial slice of reality.
1) Discretize the region into cubes of size L3in coordinate units. Discretize the Hamiltonian constraint.
To each cube xassign quantities discussed in algorithm 2.1.2.All the quantities have suffix x.The Hamiltonian
constraint is a function of near by points of x′s.To the effective Hamiltonian Hs,x add the quantum diffusion
term.
2) First wavefunction ψ({qα
x,s,∀x}, τ ) is given on the product of initial hypersurfaces Sx(τ) with normal
vector να
x(τ).Let the expectation value of the wavefunction be Qα
x(τ) in the global coordinates. The Qα
x(τ)
needs to be the origin of the coordinates qI
x,s in Sx(τ). Let < pI
x,s >be px,α in global coordinates. Let me
assume the norm of < px,α,s >is small compared that of να
x(τ).
3) Choose values for nx(τ),and evolve each the global wavefunction wavefunction ψ({qα
x,s,∀x}, τ ).To to
this at each xevolve ψ({qα
x,s,∀x}, τ ) from Qα
x(τ) along να
x(τ) to a new hypersurface Sx(τ+dτ ) with normal
να
x(τ+dτ) = mαβ
xpx,α going through point Qα
x(τ) + nx(τ)dτ να
x(τ). Set Qα
x(τ+dτ) = Qα\
x(τ) + nx(τ)dτνα(τ)
as the origin of coordinates on the new hypersurface. The Propagator and Hamiltonian for evolution was
given in equation (8) and equation (9).
4) Calculate Υ(τ),where Υ is defined by proposal three, using one of the alternative formulas discussed.
5) Change value of nx(τ). Repeat steps 3 and 4.The probability nx(τ) of values is given by exp(−crΥ)
The most probable nx(τ) is for which Υ(τ) is minimum.
6) The ψ({qα
x,s,∀x}, τ +dτ) for which Υ(τ) is minimum is the most probable new initial wavefunction.
Now start over from step 2.
If cris large enough, then the evolution happens such that Υ is minimum. To study evolution of quantum
particles in Schwarzschild or Cosmological case, the most probable nx(τ) can be chosen easily. These cases
will be later studied in this article.
2.4 Deterministic, Continuum Limit and Scale invariance
Let me assume that nature is made of large number of interacting identical discrete quantum systems at the
Planck scale. As discussed after equation (23), inclusion of discretized diffeomorphism and gauge constraints
will be discussed in future versions of this paper or in a separate paper.
The stochastic evolution in proposal three in the many body system, results in random evolution of the
system, as the classical expectation values evolve randomly. Whatever discrete quantum model one proposes
at a microscopic scale, the model need to have a proper relation to the macroscopic classical world. One
of the important aspects of this is the continuum limit. This theories have to provide smoothness and
deterministic evolution in the macroscopic limit. Another important issue is stability of the macroscopic
solution. Microscopic random fluctuations may create black holes that may consume everything around it.
We discuss this in the next update and lets focus on imposing continuum and deterministic limit.
Achieving continuum and deterministic limit for a discrete model is always a difficult problem. But
nature seems to be continuous and deterministic at macroscopic scale. One simple way to solve this problem
is given by the following proposal,which requires extensive analysis and update in future papers.
Proposal 4: Every subsystem has several mechanisms built into it explicitly such that the expectation values
of quantum variables of nearby or adjacent identical quantum systems are very close to each other. They
are such as 1) There are imaginary decay term in the action to keep the quantum variables adjacent to each
other, 2) Every system is a collection of large of subsystems each having quantum variables qI
x,s and random
variables zm, xattached to it and evolving according the first three principles, and 3) The effective variables of
18
every system is got by weighted averaging of the random and quantum variables of the underlying subsystems;
Fundamental Commutators are smoothened as a consequence of this.
In proposing this we assume that nature is fundamentally discrete and the continuum limit is due to
dynamics rather than kinematics as one would expect in continuum model. Discovering this fundamentally
discrete model is a future course of research. There many choices already available in literature, such as spin
foam models.
Let me discuss the three parts of the proposal 4 one by one. First let me discuss the first part: A simple
way to realize the first part of the proposal is to add an extra imaginary term to the action (20)
S −→ S +iX
x,s
(1
2σx(qα
y,s )|˙
Qα
x,s|)nx(τ)∆τ ,
such that σxare
1) smooth real functions of the variables ˆqβ
x,s with a lower bound,
2) functions of quantum variables at xand adjacent (or nearby) quantum systems to point x, and
3) are increasing functions as |qα
x−qα
x′|− >∞.
Now the new single-step propagator (without the quantum diffusion and global reduction) is
˜
G({qα
x, q′β
x;Sx, S′
x,∀x}; )
=1
(2π)BD Zvα
s,xpα,x >0,∀xY
x{exp(ipα,x(qα
x−q′α
x)−X
x
σx(qα
y,s)|˙qα
x,s|nx(τ)∆τ)δ(Hx)dpD
x}.
The new term with σxneed to be added to Hamiltonian Hs,x in the algorithm discussed in the last
section to enforce smoothness. The negative sign of σxin the evolution equation makes sure that |ψτ>
wavefunction weights |qα
x−qα
x′|,∀α, for every pair of quantum systems adjacent to each other in the discrete
model.
A simple choice for σxis
σx(qα
y) = X
α,Adjacent x′|qα
x−qα
x′|2.
But the problem with this function is that it will suppress the differences between qα
xand qα
x′, erasing out
the physics in long term. So alternative choices for ˆσare the following functions with minimums for non-zero
(qα
x−qα
x′)2.
1) σx(qα
y) = X
α,Adjacent x′
Aexp(B∗ |qα
x−qα
x′|2)
|qα
x−qα
x′|2,
2) σx(qα
y) = X
α,Adjacent x′
A
|qα
x−qα
x′|2+B|qα
x−qα
x′|2,
where A, B are real constants and positive. In this ˆσgoes towards infinity for both |qα
x−qα
x′|− >0and
|qα
x−qα
x′|− >∞.So this restricts the evolution of quantum state such that |qα
x−qα
x′|are finite. For large
expectation values of |qα
x|>> 0 (in quantum units), which corresponds to macroscopic case, <|qα
x−qα
x′|>
<qα
x>are
infinitesimal. This will help reproduce the continuum limit.
Another advantage of the ˆσoperator is that it limits randomness in the fields. The quantum reduction
in the last two proposals introduces randomness, and it can build to large values. The ˆσoperator can reduce
this randomness, and help fields to be smooth in the continuum limit. An extra term can be added to keep
σterms from disturbing the norm of the wavefunction.
Let me discuss the scale invariance of dynamical equations effectively. Let me study the large number
limit of the stochastic evolution equation for a many body system in a finite neighborhood Ω made of finite
19
number of systems. Let me assume the Ω is considered to made many cubes of volume ∆v, with nlattice
points in each orthonormal directions in three dimensions. Therefore Ω is made of n3cubes. Let the total
volume ∆V=n3∆v.
The evolution equation for a combined state is as follows:
|dψτ>= (X
x
αx∆V nx(τ)dt +βx
mzm
xp∆V nx(τ)dt)|ψτ>,
αx=iHx−γx−σx−βx+
mβx
m.
where xindicate different points, cis ignored, σis the operator in proposal four, and nxare set to be equal
to 1. Let me define the following averages:
¯α=Pxαx
n3,¯
βm=Pxβx
m
n3,¯zm=Pxzm
x
n3,
¯
H=PxHx
n3,¯γ=Pxγx
n3,¯σ=Pxσx
n3.
For ¯zm, we have M(¯zm¯z∗n) = 2δmn
n3.So we define ˜zm= ¯zm√n3. Then we have M(˜zm˜z∗n) = 2δmn .
Presence of σxmakes quantum amplitude <{qα
x,∀x} |ψτ>non-zero for the values of qα
xclose to each
other. Then we can approximate the quantum diffusion equation by an macroscopic averaged equation,
|dψτ>=X
x
¯α∆V nx(τ)dt +¯
βm˜zmp∆V nx(τ)dt)|ψτ>,
¯α=i¯
H−¯γ−¯σ−¯
β+
m¯
βm.
We find that the dynamical equations are scale invariant. But since the multiplying factor √∆Vof ˜zm
is larger, the system undergoes semiclassicalization rapidly in terms of the averaged values, than the each
subsystem.
Let me now focus on the commutators. Consider the scalar field φ.
φ(x) = Z(a(k) exp(ik.x) + a†(k) exp(−ik.x)) d3k
(2π)3/2√ω,
Consider the expectation value of φ(x)2in the ground state |0>:
< φ(x)2>=Z<0|a(k)a†(k′)|0>exp(−ix.(k−k′))d3kd3k′
(2π)3ω.
This needs to be calculated using the commutator of the field:
[a(k), a†(k′)] = δ(k−k′),
which is a consequence of the fundamental commutator [φ(x), π(x′)] = δ(x−x′).The result is divergent:
< φ(x)2>=Zd3k
ω=∞.
It is essential that < φ(x)2>to be finite, so that there is clear semiclassical nature for the ground state.
To achieve this commutator need to be smoothened:
20
[a(k), a†(k′)] = F(k)δ(k−k′),
resulting in
< φ(x)2>=ZF(k)d3k
ω.
If F(k) sufficiently falls of as k−>∞,< φ(x)2>becomes finite. Now the new fundamental commutator
of the field is
[φ(x), π(x′)] = f(x−x′).
where f(x) is the Fourier transform of F(k).This commutator can be achieved by considering φand π
weighted averaging of fundamental fields ˜
φ(x),˜π(x) satisfying [˜
φ(x),˜π(x′)] = δ(x−x′):
φ(x) = Z˜
φ(˜x)η(x−˜x)d˜x3,(27a)
π(x) = Z˜π(˜x)η(x−˜x)d˜x3,(27b)
where η(x) is suitable weighting function. Now we have,
[φ(x), π(x′)] = Zη(x−˜x)η(˜x−x′)d˜x3.
If we choose η(x) to be the Gaussian function 1
(d√2π)3exp(−x2
d2), we have
[φ(x), π(x′)] = exp[−(x−x′)
d2
2
],
[a(k), a†(k′)] = 1
(d√2π)3exp[−d2k2]δ(k−k′),
< φ(x)2>=4π
d2.
Now the expectation value is made finite and the commutator [φ(x), π(x′)] has been smoothened. The
equations (27a) can be written as discrete sum over large number of subsystems, to make this analysis
compatible with the second part of the fourth proposal.
3 Applications
In this section I discuss simple applications of the four principles proposed. I will first discuss the simple
minisuperspace homogenous and isotropic expanding cosmological model in which the scalar field is coupled
to