Quantum Physics, Quantum
Spacetime and Quantum Geometry
Sergio Doplicher
Universita di Roma “Sapienza”
February 16, 2016
1. Why Quantum
2. Why Quantum Spacetime
3. Quantum Minkowski Space
4. Quantum Geometry
5. QFT on QST
6. Comments on QST and Cosmology
1
In a physical theory,
the observable quantities are described by the selfad-joint elements in a C* Algebra A
(a Banach * algebra s.t. ||A∗A|| = ||A||2; equivalently auniformly closed * algebra of linear bounded operatorson a Hilbert space);
the physical states are described by states of that alge-bra
(i.e. positive normalized linear functionals)
and the expectation value of the observable A in thestate ω is given by ω(A).
3
Thus the uncertainty of A in ω is
(∆A)2 =< (A− < A > I)2 > .
If AB = BA, then A and B are compatible; otherwise,
in any state of the system,
∆A∆B & (1/2)| < AB −BA > |.
(Elementary consequence of Schwarz’s inequality for
the positive semidefinite sesquilinear form assigning to
A,B the number ω(A∗B)).
If the C* Algebra generated by the observables is com-mutative, it can be identified with C0(Ω) for some lo-cally compact space Ω, the phase space of a ClassicalTheory.
Noncommutativity of A is equivalent to the presence ofany (hence of all) the following properties:
1. There is a pair of observables which cannot be mea-sured simultaneously.
2. There is an observable A and a pure state where A
does not take a precise value (Quantum fluctuations).
3. There are pairs of distinct pure states which canbe superposed, giving rise to a pure state distinct fromboth.
4. there exists a pair of distinct pure states with a
nonzero transition probability from each to the other.
5. While time evolution in deterministic, measuring
some observable in some pure state produces a state
which is no longer pure.
Thus commutativity characterizes Classical Theories,
Noncommutatity is the sole responsible of the unintu-
itive features of Quantum Mechanics.
Is this choice a ”capriccio” or even a ”stravaganza”?
NO. It was IMPOSED on us by Nature, after nearly 30
years of misunderstandings of what Nature was telling
us.
Is QM relevant only at the atomic scale? NO:
- Superconductors (LHC);
- Specific heat of metals, ambient temperatures;
- The largest system we can think of, CMB, is a Quan-
tum System.
A word on the ”Black body problem” is here in order.
Thus classically equilibrium matter - radiation was im-
possible! Great failure of Classical Physics at the end
of the XIX Century.
CMB = Black Body radiation at T = 2.725 degrees
Kelvin, temperature fluctuations 0.0002 degree, angu-
lar size = that of a full Moon.
How did the Quantum picture of the World emerge?
** To explain the experimental curve of the spectrum
of the black body radiation we had to assume that
(Planck, 1900):
the energy of a one dimensional harmonic oscillator of
frequency ν can only take values which are integral mul-
tiples of hν (h = Planck constant).
** To explain the photoelectric effect we had to assume
that (Einstein, 1905):
Electromagnetic Radiation of frequency ν consists of an
integer number of indivisible units (photons) of energy
hν.
** To explain why matter is stable and with a discretespectrum of spectral lines (while the Planetary Model ofatoms, confirmed by the Rutherford experiment (1911),would be instable, with continuos emission / absorptionspectra according to Classical Physics!) we had to as-sume that:
Bohr (1911 - 1924): Atomic bound states have (asPlanck’s oscillator) a discrete set of energy levels, theconservation of energy is respected in jumps by emission/ absorption of a single photon:
En − Em = hνnm
This would explain the discrete nature of the observedemission / absorption spectra (hence would explain whywe see colors!), and stability of atoms.
BUT WHY?
Dramatic wreck in Classical Physics.
The answer was somehow implicit in the famous three
1905 papers of Einstein!
The photon is indivisible like a particle, but is governed
by a wave equation (Maxwell Equations)!
If the electron, as the photon, is also governed by a
wave equation, discrete energy levels arise as the dis-
crete serie of frequencies produced by a vibrating string.
Schroedinger equation.
But: a bullet shot through an open window does not
feel the window; while a wave gets diffracted going
through an opening in a dam.
Thus precision in one component of the coordinates im-plies imprecision in the same component of the speed.
And, for the electron too, the precision in one compo-
nent of the coordinates would imply imprecision in the
same component of the momentum.
Ideed:
Heisenberg (1927): assigning a point in phase space to
an electron has NO operational meaning: in any mea-
surement process which is actually at least conceptually
possible, we must have:
∆q∆p & ~
(~ = Planck constant per unit radiant).
(The three 1905 papers by Einstein included that on
Special Relativity. Key point: Operational definition of
physical quantities to criticize the notion of simultane-
ity. Heisenberg analysis took up that lesson 22 years
later!)
Can this fact be implemented by a precise mathematical
scheme? Yes:
pq − qp = −i~
(Born; Born and Jordan; Born, Heisenberg and Jordan,
1925; gives precise mathematical shape to Quantum
Conditions invented by Heisenberg).
Dirac (1926 - 1930): (Heisenberg - Born - Jordan and
Schroedinger approaches are just special representa-
tions of a new picture, where:) Observables are de-
scribed by Operators (selfadjoint operators acting on
a Hilbert space); spectrum = possible values; discrete
spectrum and continuos spectrum; for the Energy Op-
erator (Hamiltonian) the discrete spectrum describes
BOUND STATES.
The assumptions of Planck and of Bohr, stability of
atoms, discrete emission - absorption spectra, . . .
explained!
The lesson that Nature was whispering in those nearly
30 years eventually understood.
Modern view: the observables are selfadjoint elements
of a C* algebra, Hilbert spaces arise as representation
spaces (e.g. determined by a state, GNS construction).
Heart of QM: Uncertainty Relations as a shadow of
Noncommutativity.
Probabilistic aspects are a consequence.
Copernicus; Darwin; Freud; Quantum Mechanics!
LAWS OF NATURE; Naturalism; Ethics and primacy
of Culture.
Why QST
QM finitely many d. o. f.
∆q∆p & ~
positions = observables, dual to momentum;
in QFT, local observables:
A ∈ A(O);
O (double cones) - spacetime specifications, in terms of
coordinates - accessible through measurements of local
observables. Allows to formulate LOCALITY:
7
AB = BA
whenever
A ∈ A(O1), B ∈ A(O2, ),
and
O1,O2
are spacelike separated.
”Unreasonable effectiveness”.
OK at all accessible scales (down to 10−17cm., LHC atCERN); in QFT at all scales, if we neglect GRAVITA-TIONAL FORCES BETWEEN ELEMENTARY PAR-TICLES.
8
If we DON’T:
Heisenberg: localizing an event in a small region costs
energy (QM);
Einstein: energy generates a gravitational field (CGR).
A strong gravitational field can generate ”closed trapped
surfaces”.
QM + CGR:
PRINCIPLE OF Gravitational Stability against localization
of events [DFR, 1994, 95]:
The gravitational field generated by the concentra-
tion of energy required by the Heisenberg Uncer-
tainty Principle to localize an event in spacetime
should not be so strong to hide the event itself
to any distant observer - distant compared to the
Planck scale.
Spherically symmetric localization in space with accu-
racy a: an uncontrollable energy E of order 1/a has to
be transferred;
Schwarzschild radius R ' E
(in universal units where ~ = c = G = 1).
9
Hence we must have that
a & R ' 1/a;
so that
a & 1,
i.e. in CGS units
a & λP ' 1.6 · 10−33cm. (1)
(Planck length).
(FOLKLORE).
But elaborations in two significant directions are sur-
prisingly recent.
First, if we consider the energy content of a generic
quantum state where the location measurement is per-
formed, the bounds on the uncertainties should also
depend upon that energy content.
Second, if we consider generic uncertainties, the argu-
ment above suggests that they ought to be limited by
uncertainty relations.
To the first point:
background state: spherically symmetric distribution,
total energy E within a sphere of radius R, with E < R.
If we localize, in a spherically symmetric way, an event
at the origin with space accuracy a, due to the Heisen-
berg Principle the total energy will be of the order
1/a+ E.
Reasoning as above, we must then have
1
a+ E < R.
Thus, if R − E is much smaller than 1, the “minimal
distance” will be much larger than 1.
Indication that Quantum Spacetime solves the Horizon
Problem.
To the second point:
The condition that closed trapped surfaces do not arise
implies that the ∆qµ must satisfy UNCERTAINTY
RELATIONS.
Should be implemented by commutation relations,
making spacetime into a QUANTUM SPACETIME.
Dependence of Uncertainty Relations, hence of Com-
mutators between coordinates, upon background quan-
tum state i.e. upon metric tensor. Hard problem.
CGR: Geometry ∼ Dynamics
QG: Algebra ∼ Dynamics
WARNING:
Size of the known Universe: 1028cm.
Planck scale: much further away from us in log scale.
QM: no contradiction from the size of the Universe
(CMB is a quantum system!) down to scales explored
by LHC.
Below that scale no direct experimental knowledge.
GR: no contradiction from the size of the Universe down
to scales of order 10−2cm.
Below, nothing is known.
The (Planck length), 10−33cm, is very far. .
Quantum Minkowski Space
The Principle of Gravitational Stability against localization
of events implies :
∆q0 ·3∑
j=1
∆qj & 1;∑
1≤j<k≤3
∆qj∆qk & 1. (2)
[DFR 1994 - 95]; see also [TV 2012] [DMP 2013]
Must be implemented by SPACETIME commutation
relations
10
[qµ, Qν,λ] = 0; (4)
QµνQµν = 0; (5)
((1/2) [q0, . . . , q3])2 = I, (6)
where QµνQµν is a scalar, and
[q0, . . . , q3] ≡ det
q0 · · · q3... . . . ...q0 · · · q3
≡ εµνλρqµqνqλqρ =
= −(1/2)Qµν(∗Q)µν (7)
11
is a pseudoscalar, hence we use the square in the Quan-tum Conditions.
Basic model of Quantum Spacetime; implements ex-
actly Space Time Uncertainty Relations and is fullyPoincare covariant.
The classical Poincare group acts as symmetries; trans-lations, in particular, act adding to each qµ a real mul-tiple of the identity.
Gel’fand - Naimark Theorem: the classical MinkowskiSpace M is described by the commutative C* algebraof continuous functions vanishing at infinity on M ; theclassical coordinates can be viewed as commuting self-adjoint operators affiliated to that C* Algebras.
The noncommutative C* algebra of Quantum Space-
time can be associated to the above relations. The
procedure [DFR] applies to more general cases.
Assuming that the qλ, Qµν are selfadjoint operators and
that the Qµν commute strongly with one another and
with the qλ, the relations above can be seen as a bundle
of Lie Algebra relations based on the joint spectrum of
the Qµν.
Regular representations are described by representa-
tions of the C* group algebra of the unique simply con-
nected Lie group associated to the corresponding Lie
algebra.
The C* algebra of Quantum Spacetime is the C* alge-bra of a continuos field of group C* algebras based onthe spectrum of a commutative C* algebra.
In our case, that spectrum - the joint spectrum of theQµν - is the manifold Σ of the real valued antisymmetric2 - tensors fulfilling the same relations as the Qµν do: ahomogeneous space of the proper orthocronous Lorentzgroup, identified with the coset space of SL(2, C) modthe subgroup of diagonal matrices. Each of those ten-sors, can be taken to its rest frame, where the elec-tric and magnetic part are parallel unit vectors, by aboost, and go back with the inverse boost, specified bya third vector, orthogonal to those unit vectors;thus Σ can be viewed as the tangent bundle to twocopies of the unit sphere in 3 space - its base Σ1.
Irreducible representations at a point of Σ1: Shroedinger
p, q in 2 d. o. f..
The fibers, with the condition that I is not an inde-
pendent generator but is represented by I, are the C*
algebras of the Heisenberg relations in 2 degrees of free-
dom - the algebra of all compact operators on a fixed
infinite dimensional separable Hilbert space.
The continuos field can be shown to be trivial. Thus
the C* algebra E of Quantum Spacetime is identified
with the tensor product of the continuous functions
vanishing at infinity on Σ and the algebra of compact
operators.
Optimally localized states: those minimizing
Σµ(∆ωqµ)2;
minimum = 2, reached by states concentrated on Σ1,
at each point ground state of harmonic oscillator.
(Given by an optimal localization map composed with
a probability measure on Σ1).
Optimally localized states = ”points”
The classical limit of QST is M ×Σ1, i.e. M acquires
compact extra dimensions (two copies of the unit sphere
in 3 - space).
In the C* algebra E of Quantum Spacetime, define
- the von Neumann functional calculus: for each
f ∈ FL1(R4) the function f(q) of the quantum co-
ordinates qµ is given by
f(q) ≡∫f(α)eiqµα
µd4α ,
- Differential calculus: infinitesimals of translations.
- the integral over the whole space and over 3 -
space at q0 = t by∫d4qf(q) ≡
∫f(x)d4x = f(0) = Trf(q),∫
q0=tf(q)d3q ≡
∫eik0tf(k0, ~0)dk0 =
= limmTr(fm(q)∗f(q)fm(q))),
where the trace is the ordinary trace at each point ofthe joint spectrum Σ of the commutators, i.e. a Zvalued trace.
But on more general elements of our algebra both mapsgive Q - dependent results.
Important to define the interaction Hamiltonian to beused in the Gell’Mann Low formula for the S - Matrix.
Current investigations (D. Bahns, K. Fredenhagen, G.
Piacitelli, S.D.) concern models of QST where the al-
gebra of the commutators [q, q] is not central or not
even abelian.
Quantum Geometry
But to explore more thoroughly the Quantum Geometry
of Quantum Spacetime we must consider independent
events.
Quantum mechanically n independent events ought to
be described by the n − fold tensor product of E with
itself; considering arbitrary values on n we are led to
use the direct sum over all n.
If A is the C* algebra with unit over C, obtained adding
the unit to E, we will view the (n + 1) tensor power
12
Λn(A) of A over C as an A-bimodule with the productin A,
a(a1 ⊗ a2 ⊗ ...⊗ an) = (aa1)⊗ a2 ⊗ ...⊗ an;
(a1 ⊗ a2 ⊗ ...⊗ an)a = a1 ⊗ a2 ⊗ ...⊗ (ana);
and the direct sum
Λ(A) =∞⊕n=0
Λn(A)
as the A-bimodule tensor algebra,
(a1⊗a2⊗...⊗an)(b1⊗b2⊗...⊗bm) = a1⊗a2⊗...⊗(anb1)⊗b2⊗...⊗bm.
This is the natural ambient for the universal differentialcalculus, where the differential is given by
d(a0⊗· · ·⊗an) =n∑
k=0
(−1)ka0⊗· · ·⊗ak−1⊗I⊗a⊗ · · ·⊗an.
As usual d is a graded differential, i.e., if φ ∈ Λ(A), ψ ∈Λn(A), we have
d2 = 0;
d(φ · ψ) = (dφ) · ψ + (−1)nφ · dψ.
Note that A = Λ0(A) ⊂ Λ(A), and the d-stable subal-
gebra Ω(A) of Λ(A) generated by A is the universal
differential algebra. In other words, it is the subalgebra
generated by A and
da = I ⊗ a− a⊗ I
as a varies in A.
In the case of n independent events one is led to de-
scribe the spacetime coordinates of the j − th event by
qj = I ⊗ ...I ⊗⊗q⊗ I...⊗ I (q in the j - th place); in this
way, the commutator between the different spacetime
components of the qj would depend on j.
A better choice is to require that it does not; this
is achieved as follows.
The centre Z of the multiplier algebra of E is the algebra
of all bounded continuos complex functions on Σ; so
that E, and hence A, is in an obvious way a Z−bimodule.
We therefore can, and will, replace, in the definition of
Λ(A), the C - tensor product by the Z−bimodule−tensorproduct so that
dQ = 0.
As a consequence, the qj and the 2−1/2(qj − qk), j dif-ferent from k, and 2−1/2dq, obey the same space-time commutation relations, as does the normalizedbarycenter coordinates, n−1/2(q1+q2+ ...+qn); and thelatter commutes with the difference coordinates.
These facts allow us to define a quantum diagonal map
from Λn(E) to E1 (the restriction to Σ1 of E),
E(n) : E ⊗Z · · · ⊗Z E −→ E1
which factors to that restriction map and a conditionalexpectation which leaves the functions of the barycen-ter coordinates alone, and evaluates on functions of the
difference variables the universal optimally localized map(which, when composed with a probability measure onΣ1, would give the generic optimally localized state).
Replacing the classical diagonal evaluation of a functionof n arguments on Minkowski space by the quantum
diagonal map allows us to define the Quantum Wick
Product.
But working in Ω(A) as a subspace of Λ(A) allows usto use two structures:
- the tensor algebra structure described above, whereboth the A bimodule and the Z bimodule structures en-ter, essential for our reduced universal differential cal-culus;
- the pre - C* algebra structure of Λ(A), which allows
us to consider, for each element a of Λn(A), its modulus
(a∗a)1/2, its spectrum, and so on.
In particular we can study the geometric operators: sep-
aration between two independent events, area, 3 - vol-
ume, 4 - volume, given by
dq;
dq ∧ dq;
dq ∧ dq ∧ dq;
dq ∧ dq ∧ dq ∧ dq,
where, for instance, the latter is given by
V = dq ∧ dq ∧ dq ∧ dq =
εµνρσdqµdqνdqρdqσ.
Each of these forms has a number of spacetime com-
ponents:
e.g. 4 the first one (a vector), 1 the last one (a pseu-
doscalar).
Each component is a normal operator;
Theorem. For each of these forms, the sum of the
square moduli of all spacetime components is bounded
below by a multiple of the identity of unit order of
magnitude.
Although that sum is (except for the 4 - volume!)
NOT Lorentz invariant, the bound holds in any Lorentz
frame.
In particular,
- the four volume operator has pure point spectrum, atdistance 51/2 − 2 from 0;
- the Euclidean distance between two independent eventshas a lower bound of order one in Planck units.
Two distinct points can never merge to a point.
However, of course, the state where the minimum isachieved will depend upon the reference frame wherethe requirement is formulated.
(The structure of length, area and volume operators onQST has been studied in full detail [BDFP 2011]).
Thus the existence of a minimal length is not at all in
contradiction with the Lorentz covariance of the model.
A mathematical sideline, unfortunately without appli-
cations to physics (so far): for a suitable pairing, the
above differential is the Hodge dual of the Hochhschild
boundary.
To this purpose, we first define on Λ(A) an A-valued
pairing, the shuffling product, by
< a0 ⊗ ...⊗ an, b0 ⊗ ...⊗ bm >= δn,ma0b0...anbn,
so that it is a left resp right A-module map on the left
resp. right term. The restriction to Ω(A) is immedi-
ately seen to have noteworthy properties, namely:
< a0da1...dan, (db1...dbn)bn+1 >= a0[a1, b1]...[an, bn]bn+1;
< dφ, aψb >= a < dφ, ψ > b;
< ωφ, (dψ)λ >=< ω, (dψ) >< φ, λ >;
< λdψ, φω >=< λ, φ >< dψ, ω >;
with a, b ∈ A, and φ, ψ, λ, ω ∈ Ω(A).
Thus, the shuffling product induces on the universal
differential algebra a pairing which measures the non-
commutativity. In general it will be degenerate.
For any algebra with unit and with a trace, the A-valued
pairing we just described is turned into an interesting
pairing with values in the complex numbers by compo-
sition with a trace τ . Namely, let τ be a complex valued
linear map defined on a two sided ideal J, such that
τ(ab) = τ(ba), a ∈ A, b ∈ J;
τ is faithful if a in A fulfills τ(ab) = 0 for all b in J only
if a = 0.
If ΛJ(A) denotes the differential ideal in Λ(A) generated
by J, we have that ΛJ(A) is the span of elements a0 ⊗
...⊗ an where aj ∈ J for at least one j, and
< φ,ψ >∈ J, φ ∈ Λ(A), ψ ∈ ΛJ(A).
Let δ denote the Hochhschild boundary defined by
δ(a0 ⊗ ...⊗ an) =
n−1∑k=0
(−1)ka0⊗...⊗ak−1⊗akak+1⊗ak+2⊗...⊗an+(−1)nana0⊗...⊗an−1.
Then we have
Proposition. The Hochhschild boundary is a Hodge
dual of the differential for the pairing τ(< ·, · >), namely
τ(< δω, φ >) = τ(< ω, dφ >), ω, φ ∈ Λ(A).
QFT on QST
The geometry of Quantum Spacetime and the freefield theories on it are fully Poincare covariant.
One can introduce interactions in different ways, allpreserving spacetime translation and space rotation co-variance, all equivalent on ordinary Minkowski space,providing inequivalent approaches on QST; but all ofthem, sooner or later, meet problems with Lorentzcovariance, apparently due to the nontrivial action ofthe Lorentz group on the centre of the algebra of Quan-tum Spacetime.
On this points in our opinion a deeper understanding isneeded.
13
The previously described ”basic model” of QFT is taken
as a background, modifying the S matrix for polinomial
field interactions.
( Quantum Gravity would require a dynamical role of
the commutators! Beyond the horizon).
For instance, the interaction Hamiltonian on quantum
spacetime
HI(t) = λ∫q0=t
d3q : φ(q)n :
would be Q - dependent; no invariant probability
measure or mean on Σ; integrating on Σ1 [DFR 1995]
breaks Lorentz invariance.
Covariance is preserved by Yang Feldmann equations
but missed again at the level of scattering theory.
The Quantum Wick product selects a special frame
from the start. The interaction Hamiltonian on the
quantum spacetime is then given by
HI(t) = λ∫q0=t
d3q : φ(q)n :Q
where
: φn(q) :Q= E(n)(: φ(q1) · · ·φ(qn) :)
which does not depend on Q any longer, but brakes
Lorentz invariance at an earlier stage
The last mentioned approach takes into account, in the
very definition of Wick products, the fact that in our
Quantum Spacetime n (larger or equal to two) distinct
points can never merge to a point. But we can use the
canonical quantum diagonal map which associates to
functions of n independent points a function of a single
point, evaluating a conditional expectation which on
functions of the differences takes a numerical value,
associated with the minimum of the Euclidean distance
(in a given Lorentz frame!).
The “Quantum Wick Product” obtained by this
procedure leads to an interaction Hamiltonian on the
quantum spacetime given by as a constant operator–
valued function of Σ1 (i.e. HI(t) is formally in the
tensor product of C(Σ1) with the algebra of field oper-
ators).
The interaction Hamiltonian on the quantum spacetime
is then given by
HI(t) = λ∫q0=t
d3q : φ(q)n :Q
This leads to a unique prescription for the interaction
Hamiltonian on quantum spacetime. When used in the
Gell’Mann Low perturbative expansion for the S - Ma-
trix, this gives the same result as the effective non
local Hamiltonian determined by the kernel
exp
−
1
2
∑j,µ
aµj
2δ(4)
(1
n
n∑j=1
aj
).
The corresponding perturbative Gell’Mann Low formulais then free of ultraviolet divergences at each termof the perturbation expansion [BDFP 2003] .
However, those terms have a meaning only after a sortof adiabatic cutoff: the coupling constant should bechanged to a function of time, rapidly vanishing at in-finity, say depending upon a cutoff time T.
But the limit T →∞ is difficult problem, and there areindications it does not exist.
A major open problems is the following.
Suppose we apply this construction to the renormal-ized Lagrangean of a theory which is renormalizable on
the ordinary Minkowski space, with the counter terms
defined by that ordinary theory, and with finite renor-
malization constants depending upon both the Planck
length λP and the cutoff time T .
Can we find a natural dependence such that in the limit
λP → 0 and T →∞ we get back the ordinary renormal-
ized Gell-Mann Low expansion on Minkowski space?
This should depend upon a suitable way of performing
a joint limit, which hopefully yields, for the physical
value of λP , to a result which is essentially independent
of T within wide margins of variation; in that case,
that result could be taken as source of predictions to
be compared with observations.
Comments on QST and cosmology
Heuristic argument we started with: commutators be-
tween coordinates ought to depend on gµ,ν; it suggests
a scenario:
Rµ,ν − (1/2)Rgµ,ν = 8πTµ,ν(ψ);
Fg(ψ) = 0;
[qµ, qν] = iQµ,ν(g);
as said this describes a scenario, not explicitly known
equations; but should manifest some features:
14
Algebra is Dynamics.
Expect: dynamical minimal length.
In particular, divergent near singularities. Would solve
Horizon Problem, without inflationary hypothesis.
How solid are these heuristic arguments?
Exact semiclassical EE, spherically simmetric case: min-
imal length is at least λP [DMP, 2013].
Suggests:
massless scalar field semiclassical coupling with gravity;
use Quantum Wick product to define Energy - Momen-
tum Tensor Tµ,νQ (q);
Exact EE with source ω⊗ηx(Tµ,νQ (q)), where ω is a KMS
state and ηx is a state on E optimally localized at x;
these simplifying ansaetze imply a solution describing
spacetime without the horizon problem [DMP 2013].
Thus QST with a constant minimal length builds by
itself a dynamical one!
Work in progress (Morsella, Piacitelli, Pinamonti, Tomassini,
. . . , SD): search better grounds for the use of the
Quantum Wick Product on curved spacetime.
Also: possible emergence of an effective inflationary
potential as an effect of Quantum Spacetime.
Problem of the Cosmological Constant.
Dark matter.
Is Dark Matter really dark? A pale quantum - gravita-
tional moonshine might be possible.
On QST a U(1) Gauge theory becomes a U(∞) theory,
with gauge group (at each point of Σ) U(CI +K), and
neutral scalar fields are no longer gauge invariant.
Warning: Quantum effects at Planck scale result from
extrapolation of EE to that scale.
But: Newton’s law is experimentally checked only for
distances not less than .01 centimeter! (Adelberger
et al, 2003, 2004), i.e. we are extrapolating 31 steps
down in base 10 - log scale; while the size of the
known universe is ”only” 28 steps up.
15
References
Sergio Doplicher, Klaus Fredenhagen, John E. Roberts:
”The quantum structure of spacetime at the Planck
scale and quantum fields”, Commun.Math.Phys. 172
(1995) 187-220;
Luca Tomassini, Stefano Viaggiu: ”Physically moti-
vated uncertainty relations at the Planck length for an
emergent non commutative spacetime”, Class. Quan-
tum Grav. 28 (2011) 075001;
Luca Tomassini, Stefano Viaggiu: ”Building non com-
mutative spacetimes at the Planck length for Fried-
mann flat cosmologies”, arXiv:1308.2767;
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Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti:”On Quantum Spacetime and the horizon problem”, J.Geom. Phys. 74 (2013), 196-210;
Dorothea Bahns, Sergio Doplicher, Klaus Fredenhagen,Gherardo Piacitelli: ”Quantum Geometry on QuantumSpacetime: Distance, Area and Volume Operators”,Commun. Math. Phys. 308, 567-589 (2011);
Sergio Doplicher: ”The Principle of Locality. Effective-ness, fate and challenges”, Journ. Math. Phys. 2010,50th Anniversary issue;
Gherardo Piacitelli: ”Twisted Covariance as a Non In-variant Restriction of the Fully Covariant DFR Model”,Commun. Math. Phys.295:701-729,2010;
Gherardo Piacitelli: ”Non Local Theories: New Rules
for Old Diagrams”, Journal-ref: JHEP 0408 (2004)
031;
Dorothea Bahns: ”Ultraviolet Finiteness of the aver-
aged Hamiltonian on the noncommutative Minkowski
space” 2004; arXiv:hep-th/0405224.
D. Bahns , S. Doplicher, K. Fredenhagen, G. Piacitelli:
”Ultraviolet Finite Quantum Field Theory on Quantum
Spacetime”, Commun.Math.Phys.237:221-241, 2003.
Sergio Doplicher: ”Spacetime and Fields, a Quantum
Texture”, Proceedings of the 37th Karpacz Winter School
of Theoretical Physics, 2001, 204-213; arXiv:hep-th/0105251
E. G. Adelberger, B. R. Heckel and A. E. Nelson, Ann.Rev. Nucl. Part. Sci. 53, 77 (2003). arXiv:hep-ph/0307284
C.D. Hoyle, D.J. Kapner, B.R. Heckel, E.G. Adelberger,J.H. Gundlach, U. Schmidt, H.E. Swanson: ”Sub-millimeterTests of the Gravitational Inverse-square Law”, Journal-ref: Phys.Rev.D70:042004,2004 arXiv:hep-ph/0405262
A recent survey:
D. Bahns , S. Doplicher, G.Morsella, G. Piacitelli: ”Quan-tum Spacetime and Algebraic Quantum Field Theory”,arXiv:1501.03298; in ”Advances in Algebraic QuantumField Theory”, R. Brunetti, C. Dappiaggi, K. Freden-hagen, J. Yngvason, Eds. ; Springer, 2015
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