Group Cohomology of the Poincare Group and Invariant States

General Relativity and Gravitation
Group cohomology of the Poincare group and invariant quantum states
--Manuscript Draft--
Manuscript Number:
Full Title: Group cohomology of the Poincare group and invariant quantum states
Article Type: Original Research
Keywords: Noncommutative geometry; Supergravity; Quantum gravity
Corresponding Author: Charles Wang, Ph.D.
Univesity of Aberdeen
UNITED KINGDOM
Corresponding Author Secondary
Information:
Corresponding Author's Institution: Univesity of Aberdeen
Corresponding Author's Secondary
Institution:
First Author: James Moffat
First Author Secondary Information:
Order of Authors: James Moffat
Teodora Oniga
Charles Wang, Ph.D.
Order of Authors Secondary Information:
Funding Information: Carnegie Trust for the Universities of
Scotland
Ms Teodora Oniga
Engineering and Physical Sciences
Research Council
Dr Charles Wang
Abstract: Recently there has been an increasing emphasis on completing Dirac's agenda for
Quantum Field Theory through the development of completely finite theories, removing
the need for renormalisation processes. A key reason for this is the difficulty of
applying such processes to gravity which is inherently self -interacting and non-
renormalisable. Additional problems include the current need for high-levels of fine
tuning to avoid quadratic divergences in Higgs mass corrections. If Supersymmetry
applies, the mass-energies of the superpartners (if they exist) are possibly much larger
than initially thought and limit their ability to control loop divergence. This leaves open
the possibility of other approaches such as Noncommutative Geometry, incorporating
fibre bundle theory.
The aim of this paper is to consider these existing `low energy' theories incorporating
gravity beyond the Standard Model and to develop a coherent mathematical framework
based on modern quantum field theory. A number of new ideas have emerged from
this, including the role of `quantum operator algebras' and states of the algebra of
observables which are Poincare invariant.
Suggested Reviewers: Robin Tucker, Ph.D.
Professor, Lancaster University
r.tucker@lancaster.ac.uk
Prof Robin Tucker is a well-known mathematical physicist having relevant expertise
and experience in general relativity, quantum gravity and quantum field theory.
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General Relativity and Gravitation manuscript No.
(will be inserted by the editor)
Group cohomology of the Poincare group and
invariant quantum states
James Moffat · Teodora Oniga ·
Charles H.-T. Wang
Received: date / Accepted: date
Abstract Recently there has been an increasing emphasis on completing
Dirac’s agenda for Quantum Field Theory through the development of com-
pletely finite theories, removing the need for renormalisation processes. A key
reason for this is the difficulty of applying such processes to gravity which is in-
herently self -interacting and non-renormalisable. Additional problems include
the current need for high-levels of fine tuning to avoid quadratic divergences in
Higgs mass corrections. If Supersymmetry applies, the mass-energies of the su-
perpartners (if they exist) are possibly much larger than initially thought and
limit their ability to control loop divergence. This leaves open the possibility
of other approaches such as Noncommutative Geometry, incorporating fibre
bundle theory. The aim of this paper is to consider these existing ‘low energy’
theories incorporating gravity beyond the Standard Model and to develop a
coherent mathematical framework based on modern quantum field theory. A
number of new ideas have emerged from this, including the role of ‘quantum
operator algebras’ and states of the algebra of observables which are Poincare
invariant.
Keywords Noncommutative geometry · Supergravity · Quantum gravity
PACS 02.40.Gh · 04.65.+e · 04.60.-m
The authors are grateful for financial support to the Carnegie Trust for the Universities of
Scotland (TO) and the Cruickshank Trust and EPSRC GG-Top Project (CW).
J. Moffat (E-mail: jamesmoffat48@gmail.com)
T. Oniga (E-mail: t.oniga@abdn.ac.uk)
C. H.-T. Wang (E-mail: c.wang@abdn.ac.uk)
Department of Physics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, UK
Manuscript Click here to download Manuscript InvariantQuantumStates.tex
Click here to view linked References
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2 James Moffat et al.
1 Introduction
The development of perturbation methods in quantum field theory (QFT) by
Dyson, Feynman and others, was an essentially ad hoc answer to the problems
generated by the process of taking renormalisation limits for point particles.
The further development of the renormalisation group [1] based on ideas of
self-similarity at different system scales was a significant step forward. More
recently there has been an increasing emphasis on completing Dirac’s agenda
for QFT [2], [3] [4], [5], [6] through the development of completely finite the-
ories, removing the need for such renormalisation processes. A key reason for
this is the difficulty of applying such processes to gravity which is inherently
self -interacting and non-renormalisable [7]. Additional problems include the
current need for high-levels of fine tuning to avoid quadratic divergences in
Higgs mass corrections.
The aim of this paper is to consider relevant existing ‘low energy’ theories
incorporating gravity beyond the Standard Model and to develop a coherent
mathematical framework based on modern quantum field theory. Much of the
mathematical basis is derived from foundational work by one of the authors
[8], [9], [10], [11] and [12]. With a view to more general symmetry groups, we
demonstrate our present framework using the Poincare group for field theories
on a Minkowski flat background where the linearised theory of gravity coupled
with matter are applicable [13] and [14].
1.1 Supersymmetry
One way of eliminating the infinities of renormalisation is through assuming
a form of Supersymmetry (SUSY) in which additional ‘spinor charge observ-
ables’ are added to the Lie algebra of the Poincare group of spacetime sym-
metries. In an irreducible representation of the resulting ‘graded algebra’, the
centre of the algebra is reduced to multiples of the identity operator. It turns
out that all particles in this representation have equal mass, as in the Wigner
theory of irreducible representations of the Poincare group and its Lie algebra
[15]. However, the spins of the particles in this ‘super’-representation are not
fixed at a common value. In other words, the representation corresponds to a
multiplet of particles, all with the same mass, but with different spins (equal
numbers of fermions and bosons called partners and superpartners). At a given
perturbation loop level, this generates, in a na¨ıve application of the theory,
equal mass contributions to the calculation but with loop contributions of op-
posite sign which cancel, giving a finite result. In practice, this symmetry is
broken, resulting in different masses for the partners and superpartners, giving
only approximate cancellation [7]. This relates to non-commutative spacetime
and gravity through the Superfield formalism (see for example [16], [17]). It
was originated by Salam [18] involving transformations of the form:
G(xµ
, ξ, ¯ξ) = exp i(ξQ + ¯ξ ¯Q − xµ
Pµ).
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Group cohomology of the Poincare group and invariant quantum states 3
Here, coordinates xµ
identify a point in spacetime and the Grassmann variables
ξ and ¯ξ form a vector space at the point xµ
creating extreme noncommutativity
of spacetime. Q is a spinor charge observable, Pµ the generators of translations
in spacetime. These elements form a group G with a group multiplication
structure of the following form:
G(xµ
, θ, ¯θ)G(aµ
, ξ, ¯ξ) = G(xµ
+ aµ
− iξσµ ¯θ + iθσµ ¯ξ, θ + ξ, ¯θ + ¯ξ)
We can interpret this group product as shifting the locus in spacetime from
xµ
to xµ
+aµ
−iξσµ ¯θ+iθσµ ¯ξ together with additive change to the Grassmann
vector at this point.
Theories successfully describing the Electroweak and Strong forces take a
similar approach. They use a group action acting on the Lagrangian which
is symmetric, in the sense that the Lagrangian remains invariant under this
group action. However, in contrast to global SUSY, this ‘gauge group’ action
is assumed to be local in character. That is, given a point x in spacetime, the
group action is a function of x.
If we apply this approach to SUSY it is possible to create a set of global
SUSY transformations of the component fields of the Lagrangian relative to
which the Lagrangian remains invariant. However, we note that in the linear
representation of the SUSY algebra, the translation operator is represented
by ∂µ = ∂/∂xµ
. Thus it corresponds to small translations in spacetime. If
the magnitude of these translations become dependent on the local point x of
spacetime, as in a local gauge group, we have, already built into SUSY, a group
of transformations between arbitrary coordinate frames i.e. a theory of gravity.
In the literature this is referred to as Supergravity (SUGRA). We expect that
this gravitational theory should involve a spin-2 graviton and its fermionic spin
3/2 ‘gravitino’ companion particle. However, initial survey results at CERN
using the LHC ATLAS detector [19] indicate no significant deviation from the
Standard Model, implying that the mass symmetry is badly broken. In other
words, the mass-energies of the superpartners (if they exist) are possibly much
larger than initially thought and limit their ability to control loop divergence.
This leaves open the possibility of other approaches such as Noncommutative
Geometry (NCG).
1.2 Commutative algebras of observables
By comparison, a commutative operator algebra of observables is equivalent to
the set of continuous functions on a compact Hausdorff space. This equivalence
arises through the Gelfand transform of a quantum operator A, an observable
in a set of commuting observables R. This transform maps A → Â with
Â(ρ) → ρ(A) and ρ a continuous complex valued homomorphism. Since the
Gelfand transform A → Â is an algebra isomorphism, the spectrum (the set
of measurement eigenvalues) of the operator A, denoted σ(A), equals σ( Â).
If λ is an eigenvalue of Â with eigenstate ρ then ( Â − λÎ)ρ = 0. Hence
ρ(A) = λρ(I) = λ. Therefore σ(A) is a closed, compact subset S of the set:
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{ρ(A) : ρ is a complex valued continuous homomorphism} and the map ∆ :
ρ → ρ(A) is a homomorphism.
Thus if R(A, A∗
, I) denotes the commutative quantum algebra of observ-
ables generated by A, then via the Gelfand transform we can associate this
with the set C(∆−1
(S)) of continuous functions acting on the compact space
∆−1
(S), and thus with the set C(S) = C(ρ(A)). This equivalence is called the
functional calculus.
1.3 Noncommutative geometry
We have seen that the concept of non-commuting spacetime is built in to
SUSY. This potentially implies a limit on the resolution of the spacetime
variables and a corresponding limit to the energy available to the system i.e.
an upper energy or ‘UV cut-off’. NCG also considers [20] non-commutative
spacetime as part of its effort to unify quantum theory and General Relativity
(GR). There are also links to String theory at low energies [21]. If we consider
the Lagrangian L of gravity coupled to the presence of matter it can be written
as:
L = LEH + LSM .
Here LEH is the Einstein-Hilbert Lagrangian for gravity and LSM is the La-
grangian associated with the Standard Model [22]. The symmetry or ‘gauge’
group for LEH is the group of local diffeomorphisms of curvilinear reference
frames. The aim of NCG is to describe the symmetry group of LSM in a way
which is compatible with that for LEH in order to solve for the action of the
total Lagrangian L.
We take as general context for this the modern algebraic theory of rela-
tivistic quantum fields (see for example [23] in which we define the set of all
‘local observables’ O(D) which can be measured in the finite spacetime do-
main D; these are bounded linear operators whose spectrum is a subset of D.
Here we assume the availability of a flat background spacetime and a local
transformation g = Λ + a of the Poincare group P maps D → gD. If g → αg
is the corresponding mapping from P to the group of all automorphisms of
O(D) then αg(O(D)) = O(gD). We also assume that if domains D1 and D2
are disjoint (e.g. space-like separated) then O(D1) and O(D2) commute, and
that:
D1 ⊆ D2 implies O(D1) ⊆ O(D2).
The union of all the local algebras generates an algebra of all observables.
The closure in the ultraweak operator topology generates the ‘quasi-local’ von
Neumann algebra R of all observables Choosing the ultraweak topology on R
ensures that it contains an identity element.
The approach adopted by [20] and [24] consists of finding non-commutative
generalisations of the structure of LEH which can then be applied to LSM in
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order to gain greater insight into the structure of the Standard model. Our
description of the approach they take is based, in our interpretation, on the
theory of fibre bundles. A fibre bundle is a pair (E, π) where E is a topological
space and π is a projection onto a subset M called the base space. For a point
x in M , the set F = {y ∈ E ∩ π−1
(x)} is the fibre at the point x, and it is
assumed that the bundle is locally a product space about each point x. E is
a vector bundle if each fibre has the structure of a vector space. In this case
we can go further and define the fibre at x as consisting of all bases of the
vector space attached to the point x; and define the group G of changes of
basis, acting on the fibre space F. This is an example of a ‘principal bundle’.
A section or lifting s through the fibre bundle E with projection function π
mapping E to the base space M is a continuous right inverse of the function π.
In other words, s is a continuous function mapping M to E such that π(s(x)) =
x for all x in M. In NCG, commutative spacetime is replaced by a principal
product bundle X = M ×F with the fibre F a finite vector space. Commuting
functions become an algebra of non-commuting sections through this bundle.
These sections are mappings or liftings from flat spacetime M to X and a
point x ∈ M is lifted into the fibre located at x. Diffeomorphisms between
reference frames become (in the Schrodinger picture) unitary transformations
U between different bases on F, and thus (in the Heisenberg picture) operator
automorphisms of the algebra of sections. These are of the form A(x) →
UA(x)U∗
for a point x of spacetime and a section function s : x → A(x) for
all x in M. The algebraic operations of addition and multiplication are assumed
to be defined ‘fibrewise’ with each fibre thus sustaining an algebraic structure.
Thus for the fibre, we have F = {y ∈ E ∩ π−1
(x)}, (A + B)(x) = A(x) + B(x)
and (AB)(x) = A(x)B(x).
1.4 A coherent approach
In terms of a coherent quantum field theory, the implications are as follows:
(a) Each local algebra O(D) should correspond to the algebra of sections of
a principal fibre bundle with base space a finite and bounded subset of
spacetime, such as the interior of a bounded double-cone [25]. The algebraic
operations of addition and multiplication are assumed defined fibrewise for
the algebra of sections.
(b) The group of unitaries U(x) acting on the fibre space F(x) = π−1
(x)
corresponds to a local gauge group. The fibre space, in accord with NCG,
is assumed to be a separable Hilbert space of finite or at most countable
dimension.
(c) There is a limit on the resolution of the spacetime variables and a corre-
sponding limit to the energy available to the system i.e. an upper energy
or ‘UV cut-off’.
For example, if the dimension of F(x) is 2 then the gauge group is SU(2)
and a spinor ψ(x) is any element of the vector space F(x). Dropping the (x)
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notation for clarity, it is denoted ψA (A = 1, 2), and corresponds to a left-
handed Weyl 2-spinor located at the spacetime point x. Given such a spinor
ψA, a 2 × 2 matrix M acting on gives rise to another spinor ψ′
A = MB
A ψB.
Left-handed and right handed Weyl spinors are related as follows: ψA ∈ F
implies ψ∗
A = ¯ψ ˙A ∈ ˙F in the usual notation.
The representations M → M and M → (M−1
)T
are unitarily equivalent
so there is a matrix ε = εAB such that for all M, ε−1
Mε = (M−1
)T
. Matrix
ε = εAB and its inverse are given by
εAB =
0 −1
1 0
, εAB
=
0 1
−1 0
.
These matrices are used to raise and lower indices in the tensor calculus, within
which left-handed Weyl 2-spinors are covariant vectors, and right-handed Weyl
2-spinors are contravariant vectors. For example ψA
= εAB
ψB. To ensure that
such product spinor terms as ψψ = ψ2ψ1 − ψ1ψ2 are not trivially zero, we
have to ensure that the components of ψ form a Clifford algebra. Thus F has
a two-component basis, as a Hilbert space, of Clifford variables. The spinor
structure of left and right handed Weyl 2-spinors, and a gauge group including
SU(2) forms the basis of our current understanding of electro-weak unification.
The representation just discussed, consisting of the section algebra A(x)
of 2 × 2 matrix operators A(x) acting on the fibre spinor space F(x) = π−1
(x)
with fibrewise multiplication and addition, includes all such matrices and is
thus a von Neumann algebra which is both norm separable and with a trivial
centre. We assume that this applies more generally and define such an algebra
to be a ‘quantum operator algebra’, as discussed in the next section in more
detail.
1.5 Properties of quantum operator algebras
A linear associative algebra A which is also a normed linear space relative to
the norm · is a Banach *algebra if A is complete relative to the norm
topology, if AB ≤ A B for all A, B ∈ A and if there is a bijective
involution mapping (denoted by *) on A. If further we have A∗
A = A 2
for all A ∈ A, then we describe A as a C∗
-algebra.
An automorphism of A we define to be a bijective mapping from A onto
A which is an isomorphism of the algebraic structure, including the involution.
This is in some texts referred to as a ∗
automorphism.
Definition. A quantum state of A is a norm continuous linear functional
f acting on A which is real and positive on positive operators and satisfies
f(I) = 1.
To distinguish these from the more restricted vector states of the form ωx(A) =
x, Ax , in Dirac notation, we may refer to f as a generalised quantum state.
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Remark. The state space is a convex subset of the Banach dual space A∗
of
A and thus inherits the weak∗
topology from A∗
. The unit ball of A∗
is weak∗
compact and the state space is a closed and thus compact, convex subset of
the unit ball. Application of the Krein Milman Theorem [26] shows that it is
the weak∗
closed convex hull of its extreme points. These extreme points are
called pure quantum states.
A ‘normal’ state is a state which is also continuous for the ultraweak operator
topology. The ultraweak topology on the algebra A can in fact be derived from
the caracterisation of A as the dual space of the set of normal states of A, so
that, formally, (A, f) = f(A) for each element A of A and each normal state
f [27].
Definition. A C∗
algebra which is also closed in the ultraweak operator topol-
ogy is called a von Neumann algebra. A Quantum Operator Algebra is a norm
separable von Neumann algebra with trivial centre.
2 The Poincare group action on the fibre algebra
We focus now on the fibre algebra
A(x) = {A(x); A(x) is an operator acting in the Hilbert space F(x)}
which is assumed to be a Quantum Operator Algebra, and we select as our
local gauge group the Poincare group P acting as a group of automorphisms of
A(x). The explicit x-dependence should make this approach more compatible
with the left hand Lagrangian LEH.
As is well known, the Poincare group P is a Lie group and is generated
by the 3 rotations around each space axis, the 3 boosts (changes of velocity
along each space axis), and a linear translation in spacetime [28]. The finite
set of parameters associated with these, such as the angle of rotation, define
a smooth manifold, and each element g of P is of the Lie group form:
g = T (ϕ1, · · · , ϕn)
where T is the corresponding point on the manifold surface, and with n = 10
parameters in our case. The origin of the manifold T (0, 0, · · ·, 0) corresponds
to e, the group identity. We can ‘tame’ this potentially very large group by
exploiting the fact that each element g is connected to the identity. We can
consider a small neighbourhood of the identity and then extrapolate to the
whole group.
Denote ϕ = (ϕ1, · · · , ϕn) and assume ϕ is a first order infinitesimal. This
means that ϕ2
is negligible, and that T (ϕ) is close to the origin e of the group.
If U is a unitary representation of P on a Hibert space H then to first order
U(g) = U(T (ϕ)) = I + iπ(ϕµ
tµ) = I + iϕµ
π(tµ)
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with π a mapping from the Lie algebra of generators tµ of P into the set of
linear operators acting on H. U(g) a unitary operator implies that π(tµ) is a
Hermitian linear operator; an observable. Additionally, let ϕ become a second
order infinitesimal so that ϕ2
is not negligible but ϕ3
is negligible. Since U is
a group representation we have U(g1)U(g2) = U(g1g2). From this expression,
it follows that:
π[tα, tβ] = ifγ
αβπ(tγ)
with f the antisymmetric structure constants of the group. See for example
[29].
Consider now a unitary representation of the Poincare group U : g →
U(g) from P to the group of unitary operators in a quantum operator algebra
A(x). This then gives rise to a representation π of the corresponding Lie
algebra generators. In this case the centre of A(x) consists of multiples of
the identity operator. The Lie algebra of P contains both the square of the
mass-energy operator E2
and the square of the Pauli-Ljubanski operator W2
,
both of which lie in the centre of the Lie algebra. Thus π(E2
) and π(W2
) are
positive operators in the centre of A(x) and must be both positive or zero
multiples of the identity operator. When both values are non-zero we have
[28]:
π(E2
) = m2
I and π(W2
) = m2
s(s + 1)I
corresponding to a particle of rest mass m and spin s. This idea is not new but
the selection of quantum operator algebras as the building blocks of a section
algebra is new, as far as we are aware.
We now consider the translations of spacetime acting on the fibre algebra.
From [25] and our previous discussion, we know that if g is an element of the
translation subgroup T then g has the form g = exp(−iP(g)·a) where a is the
corresponding group parameter, P(g) is the Lie algebra generator of g, and
P(g) · a = −a0
P0 + a1
P1 + a2
P2 + a3
P3.
Hence under the homomorphism π between the Lie algebra and our quan-
tum operator algebra A, U(g) = exp(−iπ(P(g) · a)).
2.1 Group cohomology
A group G is called an extension of a group C by a group B if C is a normal
subgroup of G and the quotient group G/C is isomorphic to B. We assume in
what follows that the group C is abelian. These assumptions imply [30] that
the sequence:
0 → C → G → B → 0
is a short exact sequence with the mapping from C to G being the identity
map, and from G to B being (up to isomorphism) the quotient mapping from
G to G/C. The converse is also true in the sense that any short exact sequence
of groups can be identified with the extension of a group.
Given an abelian group K (say), and an arbitrary group Q, we assume that
we have mappings γ and η such that γ : Q → aut(K) is a group homomorphism
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from Q to the group of all automorphisms of K and η : Q×Q → K is a system
of factors for Q and K. Together these two mappings can be used to define a
multiplication on the set K × Q giving it a group structure:
(ξ1, y1)(ξ2, y2) = (ξ1γ(y1)(ξ2)η(y1, y2), y1y2) ∀ ξ1, ξ2 ∈ K, y1, y2 ∈ Q.
Defining
δ : (ξ, y) → y and θ : ξ → (ξ, e) with e the identity of Q
then K × Q with this group structure is an extension of K by Q, normally
denoted as KηQ. It follows that the sequence:
0 → K
θ
−→ K × Q
δ
−→ Q → 0
is a short exact sequence. If Q and K are groups with borel measure structures,
and η is borel mapping then we say that KηQ is a borel system of factors for
Q and K. Given these conditions we have the following result due to [31]:
Theorem (Mackey). There exists in the group extension KηQ a unique
locally compact topological structure relative to which KηQ is a topological
group such that the identity map from K × Q (as a product borel space) into
KηQ is a borel measurable mapping. The map θ is a bicontinuous isomor-
phism from K onto a closed normal subgroup of KηQ. The isomorphism from
KηQ/θ(K) onto Q defined by the mapping δ is also bicontinuous. Moreover,
KηQ is a separable group.
We call the theorem above Mackey’s theorem. It will be used later to
prove an important result for us when applied to the Poincare group. An early
version of this result was conjectured by one of the authors (JM) and the proof
was developed in conjunction with A. Connes [10]. Here we also introduce the
concepts of 2-cocyles and coboundaries in relation to representations of the
Poincare group which will be used later in the paper.
Let then α : g → αg be a group representation of the Poincare group P
as automorphisms of the quantum operator algebra A. If there is for each
g ∈ P a unitary Ug implementing αg then UgUh and Ugh both implement
αgh = αgαh. There is thus a phase factor λ(g, h), a complex number of mod-
ulus 1, with UgUh = λ(g, h)Ugh. By considering in a similar way the group
product ghj = (gh)j = g(hj) these phase functions λ(g, h) satisfy the require-
ment to be 2-cocycles of P.
Definition. If we set Vg = ν(g)Ug with |ν(g)| = 1 then for g ∈ P, Vg also
implements αg.
We define a lifting as the choice of an appropriate ν(g) so that g → ν(g)Ug is
a group representation i.e. the 2-cocyles are trivial. In cohomology terms the
expression
ν(g)ν(h)
ν(gh)
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defines a coboundary. A local lifting is a lifting which is restricted to a neigh-
borhood of the identity of the group.
The Poincare group P is locally Euclidean and the underlying topology
is locally compact. The Poincare group was the first locally compact non-
abelian group in which the later Mackey methodology was initially developed
by Wigner in his theory of unitary irreducible representations of P. [15], [32].
The group thus has an associated Haar measure m. The translation subgroup
T and each of the rotation groups R(xµ
) around a single spacetime axis xµ
are abelian path-connected subgroups of P.
Definition. A group representation of the Poincare group P as automor-
phisms of a quantum operator algebra A(x) is a group homomorphism α :
g → αg from P into aut(A(x)), the group of all automorphisms of A(x).
Definition. Let g → αg be a group representation of P as automorphisms of
the quantum operator algebra A(x). This representation is norm measurable
(resp. norm continuous) if, for each quantum operator A in A(x), the real val-
ued mapping g → αg −i is Haar-measurable (rep. continuous) as a function
defined on P.
Theorem 1. Let g → αg be a group representation of the Poincare group
P as automorphisms of the quantum operator algebra A(x) which is norm
measurable. Then the mapping g → αg − i is norm continuous.
Proof. Let
W = g ∈ P, αg − i ≤
ǫ
2
.
Each ∗
automorphism αg is an isometric linear map, thus if g ∈ W then
αg−1 − i = αg−1 − αid = αg(αg−1 − αid) = i − αg .
Hence, if g ∈ W then g−1
∈ W, and if g, h ∈ W, then, exploiting again the
isometry of each automorphism:
αgh − i = αg−1 (αgh − i) = αh − αg−1 ≤ αh − i + i − αg−1 ≤ ǫ.
Thus
W2
⊂ {g ∈ P; αg − i ≤ ǫ} .
Now W contains the identity id in the Lie group P and if g ∈ W , g not equal
to id, then g = exp(θK) where K is an element of the Lie Algebra. Thus W
contains intervals of the form g(θ) = {exp(θK), 0 < θ ≤ θ1} and, it follows,
must have strictly positive measure. Since P is locally compact, there is a
compact set C ⊂ W. Then [33] the set CC−1
contains a neighbourhood N of
id.
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It follows that N ⊂ CC−1
⊂ W2
⊂ {g ∈ P; αg − i ≤ ǫ} and the mapping
g → αg − i is continuous at the origin. By translation it is continuous ev-
erywhere on P.
Addendum. Although not assumed in Theorem 1 above, typically, these au-
tomorphisms αg; g ∈ P will be inner, and they will be generated by a unitary
(gauge) representation g → Wg in the fibre algebra A(x). In this case, since
A(x) is norm separable, it follows that the subset S = {Wg; g ∈ P} is also
norm separable, and we can select a countable dense subset {Wgn; n ∈ Z} of
S. Defining UW as the neighbourhood
UW = Wg; g ∈ P, Wg − I ≤
ǫ
2
of the unitary gauge group in A(x) complementary to the neighbourhood W
of the automorphism group, we can choose, for each group element g in P, a
unitary Wgn with Wg − Wgn ≤ ǫ/2. This implies that
Wgn−1g − I = Wg − Wgn ≤
ǫ
2
and thus (gn)−1
g ∈ UW and hence P = n(gn)(UW). From this, it follows
that the neighbourhood UW also has positive measure.
Definition. A unitary representation of the Poincare group P is a group ho-
momorphism U : g → Ug from P into the group of unitary operators acting
on the Hilbert space H(x).
Note that the identity of this unitary group is the identity operator I.
Definition. Let α be a group representation g → αg of P as automorphisms
of the quantum operator algebra A(x) acting on the Hilbert space H(x). A
unitary representation U : g → Ug of P implements α if αg(A) = U∗
g AUg for
all g in P and all A in A(x).
Theorem 2. Let α : g → αg be a group representation of the Poincare group
P as automorphisms of the quantum operator algebra A(x), which is norm
measurable. Then α is implemented by a norm continuous unitary represen-
tation U : g → Ug from P to the group of unitary operators acting on the
Hilbert space H(x).
Proof. If α : g → αg is a group representation of the Poincare group P as
automorphisms of the quantum operator algebra A(x), which is norm measur-
able, then by Theorem 1, α : g → αg is norm continuous. As noted earlier, the
translation subgroup T and each of the sets of boosts or rotations are abelian
path-connected subgroups. These subgroups generate P.
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Thus if gj ∈ P then let gj = g1g2g3 for example, with g1 ∈ T or R(xµ
) and
αg = αg1 αg2 αg3 . Each of these subgroups is an abelian, connected topological
group with a norm continuous representation as automorphisms of the quan-
tum operator algebra (a von Neumann algebra), it follows from [11] that each
αgi is an inner automorphism of A(x) and this inner automorphism Wgj has
a spectrum contained in the positive half plane. In fact we have [34]:
σ(Wgj ) ⊂ z; ℜz ≥
1
2
4− αgj − i
1/2
for j = 1, 2, 3.
Consider now the automorphic representation g1 → αg1 of just one of these
abelian subgroups, the translation subgroup T for example. The argument of
[11] shows that the corresponding set of unitaries {Wg1 ; g1 ∈ T } is actually a
commuting set within A(x). Let S denote the von Neumann algebra gener-
ated by {Wg1 ; g1 ∈ T }. Then S is commutative and contains the identity I of
A(x) since if id is the group identity then I = Wid. Since A(x) is an algebra
with trivial centre, S thus contains the centre Z of A(x). We showed earlier
that such a commutative quantum operator algebra is equivalent to the set of
continuous functions on a compact Hausdorff space and this equivalence arises
through the Gelfand transform A → Â with Â(ρ) = ρ(A) and ρ a continous
complex valued homomorphism. Thus we define the carrier space ΦS of S as
the set of all continuous complex valued homomorphisms on S. We then have
[36] a lifting f : ΦZ → ΦS with f(ρ)|Z = ρ for ρ ∈ ΦZ .
For ease of explanation we drop the subscript 1 for a moment here. Then, for
each g ∈ T , we can now define the Gelfand transform of a unitary Ug acting
on the carrier space ΦS of S as follows:
Ûg(ρ) =
¯ˆWg(f(ρ|Z)) ˆWg(ρ)
If we define the Gelfand transform ˆVg(ρ) =
¯ˆWg(f(ρ|Z)) then by the extended
form of the Stone-Weierstrass theorem [37], ˆVg is the Gelfand transform of
an element of the centre. Since A(x) has a trivial centre, this means that
Vg = ν(g)I with ν(g) a complex number defining a coboundary, and that
for each g ∈ G. Ug implements αg In addition, the mapping g → Ug is a
group homomorphism for if we set: Rg,h = UgUhU∗
gh then for an operator
A, Rg,hAR∗
g,h = UgUhU∗
ghAUghU∗
hU∗
g = αgαhα−1
gh (A) = A. Hence Rg,h is a
unitary in the centre. In fact Rg,h = λ(g, h)I where λ(g, h) is a 2-cocycle.
The fact that the lifting f : ΦZ → ΦS has the property f(ρ)|Z = ρ for ρ ∈ ΦZ
means that ˆRg,h(ρ) = ˆRg,h(f(ρ)) since the domain of ˆRg,h is ΦZ . But then we
have that Ûg(f(ρ)) = 1 ∀g ∈ G, thus ˆRg,h(ρ) = 1 ∀ρ ∈ ΦZ ; the 2-cocycle is
trivial. Therefore Rg,h = I ∀g, h ∈ G and g → Ug is a group representation.
Addendum. It is worth emphasizing here that the Gelfand transform ˆRg,h(ρ) =
λ(g, h)Î = 1 ∀ρ ∈ ΦZ . In other words, it is the lifting that causes the 2-cocycles
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to become trivial. This concludes the case for the abelian subgroup T .
Then if αg1 (A) = U∗
g1
AUg1 , αg2 (A) = V ∗
g2
AVg2 , αg3 (A) = W∗
g3
AWg3 we have
αg(A) = αg1 αg2 αg3 (A) = αg1 αg2 (W∗
g3
AWg3 )
= αg1 (V ∗
g2
W∗
g3
AWg3 Vg2 ) = U∗
g1
V ∗
g2
W∗
g3
AWg3 Vg2 Ug1
For g = g1g2g3 let Sg = Wg3 Vg2 Ug1 . Since S∗
g = U∗
g1
V ∗
g2
W∗
g3
we have αg(A) =
S∗
g ASg.
It remains only to prove norm continuity of the unitary representation. We
note first that unitaries preserve inner products and norms: If U is a uni-
tary operator and |x ∈ H then Ux 2
= Ux, Ux = U∗
Ux, x = x, x .
Continuing with the previous example, we have: Sg = Wg3 Vg2 Ug1 . Thus
Sg − I = Wg3 Vg2 Ug1 − I = W∗
g3
Wg3 Vg2 Ug1 − W∗
g3
from above. It fol-
lows that Sg − I = Vg2 Ug1 − W∗
g3
= Vg2 Ug1 − I + I − Wg−1
3
≤
Vg2 Ug1 − I + I − Wg−1
3
. Applying the same process again we have:
Sg − I ≤ Vg2 Ug1 − I + I − W−1
g3
≤ Ug1 − V ∗
g2
+ I − Wg−1
3
.
Thus Sg − I ≤ Ug1 − I + I − Vg−1
2
+ I − Wg−1
3
.
The result is thus proved.
2.2 Linearised gravity with matter on flat spacetime
At low energy, gravity-matter systems can be approximated using linearised
gravity with matter on flat spacetime, as recently studies in [13] and [14]. For
convenience, from now on, by spacetime we refer to the background Minkowski
flat spacetime. In a neighbourhood U of the flat spacetime point x, the fibre
bundle is locally a product bundle of the form M × F and the Lie bracket
[Pµ, Pν] = PµPν − Pν Pµ, corresponding to transport around a square ‘loop’
local to a point x in spacetime, is always zero for all µ and ν. This corresponds
to the fact that the translation subgroup of the Poincare group is abelian. Note
that if π is an algebra homomorphism from the Lie algebra of the Poincare
group then π([Pµ, Pν]) = [π(Pµ), π(Pν )]. This leads to the following definition.
Definition. The local vanishing in a neighbourhood U of spacetime point x
of the Lie bracket [Pµ, Pν] is equivalent to spacetime being (locally) flat in the
(µ, ν) directions in the neighbourhood U of spacetime point x.
Definition. A strongly continuous representation of a topological group G as
automorphisms of the quantum operator algebra A(x) is a mapping from G
to aut(A(x)) which is continuous for the strong operator topology on A(x).
We can now state and prove a result linking the flatness of spacetime via
the group cohomology of the translation subgroup of the Poincare group, to
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the properties of unitary representations of the translation subgroup of the
Poincare group.
Theorem 3. Suppose now that τ : g → τg is a strongly continuous represen-
tation of the translation subgroup T of the Poincare group as automorphisms
of the quantum operator algebra A(x) and there is a group homomorphism π
from the Lie algebra of T into A(x).
Then the representation τ : g → τg can be implemented by a strongly contin-
uous unitary representation if and only if spacetime is flat in a neighbourhood
U of spacetime point x.
Proof. We can assume that each automorphism is implemented by a unitary
operator Wg in the algebra such that τg(A) = WgAW∗
g ∀g ∈ T and A in the
algebra with Wg having the form Wg = exp(−iπ(P(g) · a)) as noted earlier.
Then it follows that {Wg; ∀g ∈ T } is a commuting set if and only if spacetime
is flat in a neighbourhood U of the point x in spacetime.
It thus remains to show the equivalence of the two following assertions:
(a) The representation τ : g → τg can be implemented by a strongly continuous
unitary representation.
(b) {Wg; ∀g ∈ T } is a commuting set of unitaries.
(a) implies (b). Since A(x) has a trivial centre, we have Wg = ν(g)Ug for a set
of scalar coboundary elements ν(g) and the result follows.
(b) implies (a).
Let N = Ker(τ) = {g ∈ T ; τg = id the identity }.
Replacing T by T/N we may assume that τg = τh implies
g = h.
We note that T is an abelian locally compact subgroup of the Poincare group
and also T is separable [38]. We may also assume [10] that the mapping g → Wg
is a borel mapping from T to the unitary group of A(x) with the weak oper-
ator topology.
Let O be the unit circle. Since A(x) is a quantum operator algebra, the uni-
tary group of the centre of A(x) is isomorphic to O. It is then easy to see that
Γ = {λWg; λ is in the unit circle, and g ∈ T } is an abelian subgroup of the
unitary group of A(x).
Define the mapping γ(λ) = λI which maps the unit circle O into the subgroup
Γ, and the mapping η(λWg) = g which maps the group Γ to the translation
group T . Then the sequence:
0 → O
γ
−→ Γ
η
−→ T → 0
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is short exact. . Thus Γ is an extension of O by T . We can in fact identify the
group Γ with the extension Oη′
T where
η′
(λ, g) = η(λWg) = g
via the map
(λ, g) → λWg.
Note that the mapping η is well defined here since we have factored out the
kernel of the representation.
The group T is a separable locally compact abelian group and is thus a stan-
dard borel space [32]. The unitary group of A(x) is also a standard borel
space, thus W−1
: Wg → g is a borel mapping [32]. η′
is thus a borel system
of factors for Oη′
T . There is thus, by Mackey’s theorem discussed earlier, a
locally compact topology on the group Γ relative to which it is a separable,
locally compact abelian group. Γ also contains the unit circle O as a normal
subgroup. With this topology on Γ, from [39] we deduce that the identity
map id : O → O lifts to a character i.e. a continuous complex valued group
homomorphism σ : Γ → C.
This lifting is sufficient to reduce the relevant 2-cocycles to triviality, resulting
in a strongly continuous unitary representation of the translation group T , as
we now show.
For each unitary Wg; g ∈ T define Ug = σ(Wg)∗
Wg. Then clearly, Ug imple-
ments the automorphism τg and σ(Ug) = 1 ∀g ∈ T . Now if
g, h ∈ T then UgUh = λ(g, h)Ugh for a 2-cocycle λ(g, h).
But then, since Ug ∈ Γ ∀g, by definition of Γ applying the group homomor-
phism σ to both sides of this relationship implies that all such 2-cocycles are
trivial, and the mapping g → Ug is a unitary group representation.
The final step is to show that this unitary representation is strongly contin-
uous. From [33], since the Hilbert space H(x) on which the unitaries act is
separable, by assumption: it is sufficient to demonstrate that the mapping
g → ξ, Ugξ is a borel measurable mapping for all g ∈ T, ξ ∈ H(x).
From the proof above we can identify the group Γ with the borel system of
factors Oη′
T .
Let the mapping J denote the identification O × T ↔ Γ ↔ Oη′
T then J is
a borel mapping by Mackey’s theorem, from our earlier discussion of group
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cohomology. Thus the mapping W : g → Wg is a borel measurable map-
ping from the translation subgroup T into the group Γ. Hence the map-
ping g → ξ, Wgξ is a measurable mapping for any ξ in the Hilbert space
H(x). In addition, the character σ : Γ → C is a continuous mapping, thus
σ ◦W : g → σ(Wg) is borel. Combining these together it follows that the map-
ping g → ξ, Ugξ = σ(Wg)∗
ξ, Wgξ is borel measurable and thus strongly
continuous.
This completes the proof of this important result, linking the flatness of space-
time and the group cohomology of the Poincare group.
3 Invariant states of the algebra of observables
In this section we change the focus to recall the overall structure of the von
Neumann algebra R of all observables, consisting of the closure of the sets of
all ‘local observables’ O(D) which can be measured in a finite and bounded
spacetime domain D such as a time bounded double cone. These then are
bounded linear operators whose spectrum is a subset of D. We assume that
if domains D1 and D2 are disjoint (e.g. space-like separated) then O(D1) and
O(D2) commute. The structure of R is much more general than that of a quan-
tum operator algebra. For example, R may well have a non-trivial centre. We
investigate the structure of R by examining its invariant states. Each section
algebra consists of operators {A ∈ O(D); A|F (x) = A(x) for all x ∈ M} which
are quantum fields, and each such section algebra is a subalgebra of R. A
Lagrangian built from a linear combination of such fields which are separately
invariant will itself be invariant. Such fields arise when the invariant state is
separating, as explained below.
Definition. Let f be a generalised quantum state of the algebra R, and
α : g → αg be a group representation of the Poincare group P as automor-
phisms of R. The state f is said to be Poincare invariant if f(αg(A)) = f(A)
for all g ∈ P and A in R. If f is a separating state for R then this implies that
αg(A) = A for all g ∈ P and A in R. Such general invariance is unlikely but
exactly the same logic applies to each of the local algebras O(D).
Definition. A quantum state of R which is invariant under the translation
subgroup T we call a symmetry state.
As discussed earlier, there is a 4-vector of Lie algebra generators of these trans-
lations aµ
denoted by the abstract vector Pµ. Acting on R, a translation aµ
maps the local algebra O(D) into the local algebra O(D + aµ
).
When we consider the translation subgroup T of the Poincare group P such
symmetry states turn out to be quite common, as we now prove. It is worth
noting here that the vacuum state of a quantum system (the state of lowest
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energy) is in general translation invariant [25] and thus a symmetry state.
Theorem 4. Let τ : g → τg be a group representation of T (the translation
subgroup of the Poincare group P) as automorphisms of the algebra of observ-
ables R which is weak∗
continuous. Then there is always at least one quantum
state of the algebra which is a symmetry state.
Proof. Let f be a state of R and define the ‘induced’ transformation νg(f) =
f ◦ τg. This means that νg(f) is also a state and νg(f)(A) = f(τg(A)).
Since the subgroup T is abelian, and the mapping g → ν(g)f is a group
homomorphism, the set {ν(g); g ∈ T } is a continuous group of commuting
transformations of the dual space R∗
. If f is a quantum state of the algebra
then define E to be the weak∗
closed convex hull of the set {ν(g)f; g ∈ T }.
Then E is a weak∗
compact convex set and each ν(g) : E → E . By the
Markov-Kakutani fixed point theorem [40] it follows that E has an invariant
element. Thus each quantum state generates a symmetry state which is in
the closed convex hull of the set of all translations of f. By translations we
mean of course the automorphic representation of these translations acting on
observables in our algebra.
4 Concluding remarks
Motivated by Supersymmetry, Supergravity and Noncommutative Geometry,
we have presented an analysis of the group cohomology of the Poincare group
and the associated invariant quantum states. In a future paper, we aim to
consider the relationship between noncommutative ergodic theory and their
symmetry states. We will also consider a ‘Moyal type’ deformation of nor-
mal projection operator equivalence, which we call a P-twisted equivalence
relation.
The used of the Poincare group in the present work arises from the available
flatness of the background spacetime in linearised gravity. Nonetheless, it will
be of interest to further apply our framework to generic curved spacetime as
in full General Relativity with gravitational and matter fields involving diffeo-
morphism and gauge symmetries of local nature. The details of such extended
investigations and resulting invariant states relevant for the non-perturbative
quantisation of gravity and gauge theories [41], [42] will be reported elsewhere.
References
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Group Cohomology of the Poincare Group and Invariant States

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