Possible polarisation and spin dependent aspects of quantum gravity
Possible polarisation and spin dependent aspects
of quantum gravity
D. V. Ahluwalia-Khalilova, N. G. Gresnigt, Alex B. Nielsen,
D. Schritt, T. F. Watson
Department of Physics and Astronomy, Rutherford Building
University of Canterbury
Private Bag 4800, Christchurch 8020, New Zealand
E-mail: dharamvir.ahluwalia-khalilova@canterbury.ac.nz
We argue that quantum gravity theories that carry a Lie algebraic modification
of the Poincaré and Heisenberg algebras inevitably provide inhomogeneities that
may serve as seeds for cosmological structure formation. Furthermore, in this class
of theories one must expect a strong polarisation and spin dependence of various
quantum-gravity effects.
I. Introduction— Quantum gravity proposals often come with a modification of the
Heisenberg, and Poincaré, algebras. Confining ourselves to Lie algebraic modifications,
we argue that the underlying physical space of all such theories must be inhomogeneous.
In order to establish this result, we first review how, within a quantum framework, the
homogeneity and continuity of physical space lead inevitably to the Heisenberg algebra.
We then review general arguments that hint towards algebraic modifications encountered
in quantum gravity proposals. Next, we argue that a natural extension of physical laws
to the Planck scale can be obtained by a Lie algebraic modification of the Poincaré and
Heisenberg algebras in such a way that the resulting algebra is immune to infinitesimal
perturbations in its structure constants. With the context so chosen, we establish the
main thesis: that quantum gravity theories of the aforementioned class inevitably provide
inhomogeneities that may serve as seeds for structure formation; and that quantum
gravity induced effects may carry a strong polarisation and spin dependence.
The established results are not restricted to the chosen algebra but may easily be
extended to all Lie algebraic modifications that alter the Heisenberg algebra.1
2. Homogeneity and continuity of physical space, and its imprint in the Heisenberg
algebra— In order to understand the fundamental origin of primordial inhomogeneities
we will first review the fundamental connection between the homogeneity and continuity
of physical space and the Heisenberg algebra. It is in this spirit that we remind our
reader of an argument that is presented, for example, by Isham in [1, Section 7.2.2].
There it is shown that, in the general quantum mechanical framework, and under the
following two assumptions,
— physical space is homogeneous,
— any spatial distance r can be divided in to two equal parts, r = r/2 + r/2,
it necessarily follows that the operator x associated with position measurements along
the x-axis, and the generator of displacements dx along the x-direction, satisfy [x, dx] = i.
If one now requires consistency with the elementary wave mechanics of Heisenberg,
one must identify dx with px/h̄ (px is the operator associated with momentum mea-
surements along the x-direction). This gives, [x, px] = ih̄. Without any additional
assumptions, the argument easily generalises to yield the entire Heisenberg algebra
[xj, pk] = ih̄δjk, [pj, pk] = 0, [xj, xk] = 0, where xj, j = 1, 2, 3, are the position
operators associated with the three coordinate axes, where the observer is assumed to
be located at the origin of the coordinate system.
Thus it is evident that a quantum description of physical reality, with spatial homo-
geneity and continuity, inevitably leads to the Heisenberg algebra.
3. On the need to go beyond the Heisenberg and Poincaré algebraic-based description
of physical reality— From an algebraic point of view much of the success of modern
physics can be traced back to the Poincaré and Heisenberg algebras. Had the latter
algebra been discovered before the former, the conceptual formulation and evolution of
theoretical physics would have been significantly different. For instance, it is a direct
implication of Heisenberg’s fundamental commutator [xi, pj] = ih̄δij (with i, j = 1, 2, 3),
that events should be characterised not only by their spatiotemporal location xµ, but also
1A slightly weaker argument can be constructed for non-Lie algebraic proposals when we confine
ourselves to probed distances significantly larger than the length scale associated with the loss of spatial
continuity.
by the associated energy momentum pµ; and that should be done in a manner consistent
with the fundamental measurement uncertainties inherent in the formalism. The reader
may wish to come back to these remarks in the context of Eq. (16) where one shall
find that in a specific sense the physical space that underlies the conformal algebra does
indeed combine the notions of spacetime and energy momentum. Furthermore, as will
be seen from Eq. (18) and the subsequent remarks, this interplay becomes increasingly
important as we consider the early universe above ≈ 100 GeV.
In the mentioned description the interplay of the general relativistic and quantum
mechanical frameworks becomes inseparably bound. To see this, consider the well-known
thought experiment to probe spacetime at spatial resolutions around the Planck length
h̄G/c3. If one does that, one ends up creating a Planck mass mP
h̄c/G
black hole. This fleeting structure carries a temperature T ≈ 1030K and evaporates
in a thermal explosion in ≈ 10−40 seconds. This, incidentally, is a long time – about
ten thousand fold the Planck time τP
h̄G/c5. The formation and evaporation of
the black hole places a fundamental limit on the spatiotemporal resolution with which
spacetime can be probed.
The authors of [2, 3] have argued that once gravitational effects associated with
the quantum measurement process are accounted for, the Heisenberg algebra, and in
particular the commutator [xj, pk], must be modified. The role of gravity in the quantum
measurement process was also emphasised by Penrose [4].
From the above discussion, we take it as suggestive that an operationally-
defined view of physical space (or, its generalisation) shall inevitably ask for
the length scale, `P to play an important role.
In the context of the continuity of physical space we will take it as a working hypothesis
that, just as a lack of commutativity of the x and px operators does not render the
associated eigenvalues discrete, similarly the existence of a non-vanishing `P does not
necessarily make the underlying space lose its continuum nature. This is a highly non-
trivial issue requiring a detailed discussion from which we here refrain; yet, an element
of justification shall become apparent below.
From a dynamical point of view, as early as late 1800’s, the symmetries of Maxwell’s
equations were already suggesting a merger of space and time into one physical entity,
spacetime [5]. Algebraically, these symmetries are encoded in the Poincaré algebra. The
emergent unification of space and time called for a new fundamental invariant, c, the
speed of light (already contained in Maxwell’s equations). From an empirical point
of view, the Michelson-Morley experiment established the constancy of the speed of
light for all inertial observers, and thus re-confirmed, in the Einsteinian framework, the
implications of the Poincaré spacetime symmetries.
Concurrently, we note that while in classical statistical mechanics it is the volume that
determines the number of accessible states and hence the entropy, the situation is dramat-
ically different in a gravito-quantum mechanical setting. One example of this assertion
may be found in the well-known Bekenstein-Hawking entropy result for a Schwarzschild
black hole, SBH = (k/4)(A/`
P ); where k is the Boltzmann constant, and A is the surface
area of the sphere contained within the event horizon of the black hole. Thus quan-
tum mechanical and gravitational realms conspire to suggest the holographic conjecture
[6, 7, 8]. The underlying physics is perhaps two fold: (a) contributions from higher
momenta in quantum fields to the number of accessible states is dramatically reduced
because these are screened by the associated event horizons; and (b) the accessible states
for a quantum system are severely influenced by the behaviour of the wave function at
the boundary.
From this discussion, we take it as suggestive that in quantum cosmology/gravity
the new operationally-defined view of physical space shall inevitably ask for a
cosmological length scale, `C.
These observations prepare us to reach the next trail in our essay.
In the immediate aftermath of cosmic creation with the big bang, the physical reality
knew of no inertial frames of Einstein. This is due to the fact that massive particles had
yet to appear on the scene. The spacetime symmetries at cosmic creation are encoded in
the conformal algebra. So, whatever new operational view of spacetime emerges, it must
somehow also incorporate a process by which one evolves from the “conformal phase” of
the universe at cosmic creation to the present (see Fig. 1).
Algebraically, we take it to suggest that there must be a mechanism that
describes how the present day Poincaré-algebraic description relates to the
conformal-algebraic description of the universe at its birth.
We parenthetically note that in the conformal phase, where leptons and quarks were
yet to acquire mass (through the Higg’s mechanism, or something of that nature), the
operationally-accessible symmetries are not Poincaré but conformal. This is so because
to define rest frames, so essential for operationally establishing the Poincaré algebra, one
needs massive particles. In the transition when massive particles come to exist, the local
algebraic symmetries of general relativity suffer an operational change. Consequently,
for the cosmic epoch before ≈ 100 GeV general relativistic description of physical reality
might require modification.
4. A new algebra for quantum gravity and the emergent inhomogeneity of physical
space— Mathematically, a Lie algebra incorporating the three italicised items in Sec. 3
already exists. It was inspired by Faddeev’s mathematical analysis of the quantum and
relativistic revolutions of the last century [9] and was followed up by Vilela Mendes in
his 1994 paper [10]. The uniqueness of the said algebra was then explored through a
Lie-algebraic investigation of its stability by Chryssomalakos and Okon, in 2004 [11].
Some of the physical implications were subsequently explored in Refs. [12, 13], and its
Clifford-algebraic representation was provided by Gresnigt et al. [14]. Its importance
was further noted in CERN Courier [15].
However, its candidacy for the algebra underlying quantum cosmology/gravity has
been difficult to assert. This is essentially due to a perplexing observation made in
Ref. [11] regarding the interpretation of the operators associated with the spacetime
events. In this essay we overcome this interpretational hurdle and argue that it contains
all the desired features for such an algebra.
To this end we first write down what has come to be known as the Stabilised Poincaré-
Heisenberg Algebra (SPHA) and then proceed with the interpretational issues. The
SPHA contains the Lorentz sector (we follow the widespread physics convention which
takes the Jµν as dimensionless and Pν as dimensionful)
[Jµν ,Jρσ] = i (ηνρJµσ + ηµσJνρ − ηµρJνσ − ηνσJµρ) (1)
This remains unchanged (as is strongly suggested by the analysis presented in [16]), as
does the commutator
[Jµν ,Pλ] = i (ηνλPµ − ηµλPν) (2)
These are supplemented by the following modified sector
[Jµν ,Xλ] = i (ηνλXµ − ηµλXν) (3)
[Pµ,Pν ] = iqα1Jµν (4)
[Xµ,Xν ] = iqα2Jµν (5)
[Pµ,Xν ] = iqηµνI + iqα3 Jµν (6)
[Pµ, I] = iα1Xµ − iα3Pµ (7)
[Xµ, I] = iα3Xµ − iα2Pµ (8)
[Jµν , I] = 0 (9)
The metric ηµν is taken to have the signature (1,−1,−1,−1). The SPHA is stable,
except for the instability surface defined by α23 = α1α2 (see Fig. 2). Away from the
instability surface the SPHA is immune to infinitesimal perturbations in its structure
constants. This distinguishes SPHA from many of the competing algebraic structures
because a physical theory based on such an algebra is likely to be free from “fine tuning”
problems. This is essentially self evident because if an algebraic structure does not carry
this immunity, one can hardly expect the physical theory based upon such an algebra to
enjoy the opposite.
The SPHA involves three parameters α1, α2, α3. The c and h̄ arise in the process of
the Lie algebraic stabilisation that takes us from the Galilean relativity to Einsteinian
relativity, and from classical mechanics to quantum mechanics. Their specific values are
fixed by experiment. Similarly, α1, α2, α3 owe their origin to a similar stabilisation of the
combined Poincaré and Heisenberg algebra.
Except for the fact that α1 must be a measure of the size of the observable universe
(here assumed to be operationally determined from the Hubble parameter), the Lie
algebraic procedure for obtaining SPHA does not determine α1, α2, α3. Dimensional and
phenomenological considerations, along with the requirement that we obtain physically
viable limits, suggest the following identifications:2
α1 :=
where `C is of the order of the Hubble radius, and therefore it depends on the cosmic
epoch. The introductory remarks, and existing data suggest that [11]
In the limit `P → 0, `C → ∞, β → 0, I → I, the identity operator, the SPHA splits
into Heisenberg and Poincaré algebras. In that limit, the symbols Xµ → xµ,Pµ →
pµ,Jµν → Jµν , and I → I. Thus xµ, pµ, Jµν , I acquire their traditional meaning, while
2In making the identifications it is understood that these may be true up to a multiplicative factor
of the order of unity.
Xµ,Pµ,Jµν , I are to be considered their generalisations. In particular xµ should then be
interpreted as the generator of energy-momentum translation. The latter parallels the
canonical interpretation of pµ as the generator of spacetime translation. This interpre-
tation, we believe, removes the problematic interpretational aspects associated with Xµ
in the analysis of Ref. [11].
The identification of q with h̄ is dictated by the demand that we recover the Heisen-
berg algebra. It also suggests that at the present cosmic epoch α3 should not allow the
second term in the right hand side of equation (6) to have a significant contribution. It
will become apparent below that α3 is intricately connected to the conformal algebraic
limit of SPHA. With these identifications, and with α3 renamed as the dimensionless
parameter β, the SPHA takes the form
[Jµν ,Jρσ] = i (ηνρJµσ + ηµσJνρ − ηµρJνσ − ηνσJµρ) (12)
[Jµν ,Pλ] = i (ηνλPµ − ηµλPν) , [Jµν ,Xλ] = i (ηνλXµ − ηµλXν) (13)
[Pµ,Pν ] = i
h̄2/`2C
Jµν , [Xµ,Xν ] = i`2PJµν , [Pµ,Xν ] = ih̄ηµνI + ih̄β Jµν (14)
[Pµ, I] = i
h̄/`2C
Xµ − iβPµ, [Xµ, I] = iβXµ − i
`2P/h̄
Pµ, [Jµν , I] = 0 (15)
Since cosmic creation began with massless particles, it should be encouraging if in
some limit SPHA reduced to the conformal algebra. This is indeed the case. It follows
from a somewhat lengthy, though simple, exercise. Towards examining this question we
introduce two new operators
P̃µ = aPµ + bXµ, X̃µ = a′Xµ + b′Pµ (16)
and find that if the introduced parameters a, b, a′, b′ satisfy the the following conditions
, b =
, a′ =
`2C(1− β)
with β2 restricted to the value 1 + (`2P/`
C), then SPHA written in terms of P̃µ and X̃µ
satisfies the conformal algebra [17, Sec. 4.1].
Using these results, we can re-express P̃µ and X̃µ in a fashion that supports the view
taken in the opening paragraph of this section
P̃µ = a
(1− β)Xµ
, X̃µ = a′
(1− β)Pµ
β2 = 1 +
. (19)
Near the big bang, `C ≈ `P and thus β → ±
2 (see, Fig. 1). This results in a significant
mixing of the Xµ and Pµ in the conformal algebraic description in terms of X̃µ and P̃µ.
In contrast, hypothetically, had we been on the conformal surface at present then
taking `C � `P makes β → ±1. Consequently, for β → +1, P̃µ becomes identical to
Pµ up to a multiplicative scale factor a. Similarly, X̃µ becomes identical to Xµ up to a
multiplicative scale factor a′. As is evident from Eq. (17), the multiplicative scale factors
a and a′ are constrained by the relation aa′ = `2P/(`
C(1− β)). We expect that similar
modifications to spacetime symmetries would occur if we were to explore it at Planckian
energies in the present epoch. For β → −1 ( `C � `P ), one again obtains significant
mixing of the Xµ and Pµ.
By containing `P and `C , the SPHA unifies the extreme microscopic with the extreme
macroscopic, i.e., the cosmological. In the early universe it allows for the existence of
conformal symmetry. The significant departure from the Heisenberg algebra at big bang,
yields primordial inhomogeneities in the underlying physical space and the quantum fields
that it supports. The latter is an unavoidable consequence of the discussion presented
in Sec. 2.3
5. Polarisation and spin dependence of the cosmic inhomogeneities and other quantum
gravity effects— A careful examination of SPHA presented in equations (12-15) reveals
a strong Jµν dependence of the modifications to the Heisenberg algebra. Physically, this
translates to the following representative implications
— The induced primordial cosmic inhomogeneities are dependent on spin and polari-
sation of the fields for which these are calculated.
— The operationally-inferred commutativity/non-commutativity of the physical space
depends on the spin and polarisation of the probing particle.
— The just enumerated observation implies that a violation of equivalence principle
is inherent in the SPHA based quantum gravity.
3Any one of the other suggestions in quantum gravity that modify the Heisenberg algebra (see, e.g.,
references [18]-[28]) carry similar implications for homogeneity and isotropy of the physical space.
— Since Heisenberg algebra uniquely determines the nature of the wave particle du-
ality [27, 28] (including the de Broglie result “λ = h/p”), it would undergo spin
and polarisation dependent changes in quantum gravity based on SPHA.
All these results carry over to any theory of quantum gravity that modifies the Heisenberg
algebra with a Jµν dependence.
6. Conclusion— In this essay we have motivated a new candidate for the algebra which
may underlie a physically viable and consistent theory of quantum cosmology/gravity.
Besides yielding an algebraic unification of the extreme microscopic and cosmological
scales, it generalises the notion of conformal symmetry. The modifications to the Heisen-
berg algebra at the present cosmic epoch are negligibly small; but when `C and `P are of
the same order (i.e, at, and near, the big bang), the induced inhomogeneities are intrinsic
to the nature of physical space. These can then be amplified by the cosmic evolution
and result in important back reaction effects [29, 30, 31, 32]. An important aspect of
the SPHA-based quantum gravity is that it inevitably provides inhomogeneities that
may serve as an important ingredient for structure formation [33]. Furthermore, in this
class of theories one must expect a strong polarisation and spin dependence of various
quantum-gravity effects.
Acknowledgements— We wish to thank Daniel Grumiller and Peter West for their in-
sightful questions and suggestions.
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Possible Scenario 1
Possible Scenario 2
Big bang Present FutureFuture
Instability
Conformal
�!!!2
Figure 1: This figure is a cut, at `P = 1 (with h̄ set to unity), of Fig. 2 and it schemat-
ically shows the cosmic evolution along two possible scenarios. For this purpose, only
β ≥ 0 values have been taken. The β < 0 sector can easily be inferred from symmetry
consideration. In one of the scenarios the conformal symmetry of the early universe is
lost without crossing the instability surface, while in the other it crosses that surface. In
the latter case the algebra changes [11] from so(2, 4) to so(1, 5). This crossover, we spec-
ulate, may be related to the mass-generating process of spontaneous symmetry breaking
(SSB) of the standard model of high energy physics. The big bang is here identified with
`C ≈ `P .
�����������{c2
Instability Cone
Conformal Surface
Poincaré-Heisenberg
Algebra HunstableL
Figure 2: The unmarked arrow is the `2P (= h̄α2) axis. The Poincaré-Heisenberg algebra
corresponds to the origin of the parameters space, which coincides with the apex of the
instability cone. In reference to Eq. (10), note that `2C = h̄/α1. Here, β is a dimensionless
parameter that corresponds to a generalisation of the conformal algebra. The SPHA lives
in the entire (`C , `P , β) space except for the surface of instability. The SPHA becomes
conformal for all values of (`C , `P , β) that lie on the “conformal surface”.
