Relativity principle without space-time

Relativity principle without space-time
Giacomo Mauro D'Ariano Università degli Studi di Pavia
Is quantum theory exact?
The endeavor for the theory beyond standard quantum mechanics.
Second Edition FQT2015
Frascati September 23-25 2015
Program
Derive the whole Physics from principles
Physics as an axiomatic theory
with thorough physical interpretation
Principles for
Quantum Theory
Principles for
Mechanics
Selected for a Viewpoint in Physics
PHYSICAL REVIEW A 84, 012311 (2011)
Informational derivation of quantum theory
Giulio Chiribella∗
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Ontario, Canada N2L 2Y5†
Paolo Perinotti
Alessandro Bisio
Alessandro Tosini
Marco Erba
Franco Manessi
Nicola Mosco
Giacomo Mauro D’Ariano‡ and Paolo Perinotti§
QUIT Group, Dipartimento di Fisica “A. Volta” and INFN Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy∥
(Received 29 November 2010; published 11 July 2011)
We derive quantum theory from purely informational principles. Five elementary axioms—causality, perfect
distinguishability, ideal compression, local distinguishability, and pure conditioning—define a broad class of
theories of information processing that can be regarded as standard. One postulate—purification—singles out
quantum theory within this class.
DOI: 10.1103/PhysRevA.84.012311
PACS number(s): 03.67.Ac, 03.65.Ta
very beginning. Von Neumann himself expressed his
Principles for Quantum the
Theory
dissatisfaction with his mathematical formulation of quan-
I. INTRODUCTION
han 80 years after its formulation, quantum theory
sterious. The theory has a solid mathematical founddressed by Hilbert, von Neumann, and Nordheim
1] and brought to completion in the monumental
on Neumann [2]. However, this formulation is based
stract framework of Hilbert spaces and self-adjoint
which, to say the least, are far from having an
physical meaning. For example, the postulate stating
ure states of a physical system are represented by
ors in a suitable Hilbert space appears as rather
which are the physical laws that lead to this very
hoice of mathematical representation? The problem
standard textbook formulations of quantum theory
e postulates therein impose particular mathematical
without providing any fundamental reason for this
e mathematics of Hilbert spaces is adopted without
uestioning as a prescription that “works well” when
black box to produce experimental predictions. In
tory axiomatization of quantum theory, instead, the
ical structures of Hilbert spaces (or C* algebras)
merge as consequences of physically meaningful
, that is, postulates formulated exclusively in the
of physics: this language refers to notions like
P1. Causality
tum theory with the surprising words “I don’t believe in
Hilbert space anymore,” reported by Birkhoff in [3]. Realizing the physical relevance of the axiomatization problem, Birkhoff and von Neumann made an attempt to understand quantum theory as a new form of logic [4]:
the key idea was that propositions about the physical world
must be treated in a suitable logical framework, different from
classical logics, where the operations AND and OR are no longer
distributive. This work inaugurated the tradition of quantum
logics, which led to several attempts to axiomatize quantum
theory, notably by Mackey [5] and Jauch and Piron [6] (see
Ref. [7] for a review on the more recent progresses of quantum
logics). In general, a certain degree of technicality, mainly
related to the emphasis on infinite-dimensional systems, makes
these results far from providing a clear-cut description of
quantum theory in terms of fundamental principles. Later
Ludwig initiated an axiomatization program [8] adopting an
operational approach, where the basic notions are those of
preparation devices and measuring devices and the postulates
specify how preparations and measurements combine to give
the probabilities of experimental outcomes. However, despite
the original intent, Ludwig’s axiomatization did not succeed
P2. Local discriminability
P3. Purification
• Mechanics (QFT) derived in terms of
countably many quantum systems in
interaction
add principles
P4. Atomicity of composition
P5. Perfect distinguishability
Min algorithmic complexity principle
P6. Lossless Compressibility
•
Book from CUP soon!
•
•
homogeneity
locality
reversibility
Principles for
Quantum Field Theory
• QFT derived in terms of countably
many quantum systems in interaction
G
g
add principles
Min algorithmic complexity principle
•
•
•
•
•
•
•
homogeneity
locality
reversibility
}
g’
h
Quantum Cellular
Automata on the
Cayley graph of a
group G
}
linearity
isotropy
minimal-dimension
Cayley qi-embedded in Rd
Restrictions
G=<h1,h2,…|r1,r2,…>=:<S+|R>
Linearity (free QFT)
Principles for
Quantum Field Theory
Quantum Cellular Automaton
• QFT derived in terms of countably
many quantum systems in interaction
G
Min algorithmic complexity principle
•
•
•
•
•
•
•
homogeneity
locality
reversibility
}
Quantum Cellular
Automata on the
Cayley graph of a
group G
}
linearity
isotropy
minimal-dimension
Cayley qi-embedded in Rd
G virtually Abelian
Restrictions
(geometric group theory)
†
U U =A
von Neumann algebra
add principles
Quantum Walk
g
Fock space
Isotropy
There exists a
g’group L ofh
permutations of S+, transitive over
S+ that leaves the Cayley graph
invariant
Quantum
(free QFT)
• Linearity
a nontrivial
unitary Walk
s-dimensional
(projective)
representation
Quantum
Cellular
Automaton {Ll} of L
such that:
•
†
U
U
=
A
G=<h
,…>=:<S+|R>
!
1,h2,…|r1,r2!
A=
T h ⊗ Ah =
Tlh ⊗ Ll Ah L†l
Fock space von Neumann algebra
h∈S
h∈S
a
b
G = ⟨a, b|aba
−1 −1
b
⟩≡Z ×Z
FIG.
NOT
Theorem: every virtually Abelian QW
with cell dimension s is equivalent to
an Abelian QW with quantum cell
dimension2multiple
of
s.
G. 3. Cayley graph of G = ha, b|a b 2 i. The graph is
b
a
Quantum walk on Cayley graph
Theorem: A group is quasi-isometrically
embeddable in Rd iff it is virtually Abelian
Virtually Abelian groups
have polynomial growth
(Gromov)
# points ~rd
• G hyperbolic
raph is
5
4
2
FIG. 4. Cayley graph of G = ha, b|a , b , (ab) i.
• G hyperbolic → exponential growth
raph is
# points ~ exp(r)
information “transmitted” over
the graph decreases
exp(-r)
FIG. 4.as Cayley
5
4
2
graph of G = ha, b|a , b , (ab) i.
• Relativistic regime (k≪1): free QFT
(Weyl, Dirac, and Maxwell)
Informationalism:
Principles for QFT
• Ultra-relativistic regime (k~1) [Planck scale]:
nonlinear Lorentz
• QFT derived in terms of countably
many quantum systems in interaction
add principles
Min algorithmic complexity principle
•
•
•
•
•
•
•
homogeneity
locality
reversibility
}
Quantum Cellular
Automata on the
Cayley graph of a
group G
}
linearity
isotropy
minimal-dimension
Cayley qi-embedded in Rd
G virtually Abelian
Restrictions
• QFT derived:
•
without assuming Special Relativity
•
without assuming mechanics (quantum ab-initio)
• QCA is a discrete theory
Motivations to keep it discrete:
1. Discrete contains continuum as special regime
2. Testing mechanisms in quantum simulations
3. Falsifiable discrete-scale hypothesis
4. Natural scenario for holographic principle
5. Solves all issues in QFT originating from
continuum:
i) uv divergencies
ii) localization issue
iii) Path-integral
6. Fully-fledged theory to evaluate cutoffs
D'Ariano, Perinotti, PRA 90 062106 (2014)
The Weyl QCA
☞ Minimal dimension for nontrivial unitary Abelian QW is s=2
Qi-embeddability in R3
• Uni+Iso the only possible Cayley is the BCC!!
• Iso
B
Fermionic ψ (d=3)
Unitary operator:
A=
±
Ak
Two QWs
connected
by P
Z
=
dk Ak
B
i
x (sx cy cz
⌥i
y (cx sy cz
i
z (cx cy sz
± cx sy sz )
⌥ sx cy sz )
± sx sy cz )
+ I(cx cy cz ⌥ sx sy sz )
kα
sα = sin √
3
kα
cα = cos √
3
D'Ariano, Perinotti, PRA 90 062106 (2014)
The Weyl QCA
i∂t ψ(t) ≃ 2i [ψ(t + 1) − ψ(t − 1)] = 2i (A − A† )ψ(t)
±
i
(A
k
2
†
A±
k )=+
x (sx cy cz
±
y (cx sy cz
+
k⌧1
Two QCAs
connected
by P
z (cx cy sz
± cx sy sz )
“Hamiltonian”
⌥ sx cy sz )
± sx sy cz )
i@t =
p1
3
±
Ak
=
±
·k
☜ Weyl equation!
i
x (sx cy cz
⌥i
y (cx sy cz
i
z (cx cy sz
± cx sy sz )
⌥ sx cy sz )
± sx sy cz )
+ I(cx cy cz ⌥ sx sy sz )
±
:= (
x, ± y ,
kα
sα = sin √
3
kα
cα = cos √
3
z)
D'Ariano, Perinotti, PRA 90 062106 (2014)
Bisio, D'Ariano, Perinotti, arXiv:1407.6928
Dirac QCA
Maxwell QCA
⌦
Local coupling:Ak coupled with its inverse
with off-diagonal identity block matrix
Ek±
=
!
nA±
k
imI
n 2 + m2 = 1
imI
±†
nAk
"
1
[n(cx cy cz ⌥ sx sy sz )]
Dirac in relativistic limit
F (k) =
dq
f (q) ˜( k2
2⇡
q)
µ
'( k2 + q)
Maxwell in relativistic limit k ⌧ 1
Boson: emergent from entangled Fermions
(De Broglie neutrino-theory of photon)
±
Ek CPT-connected!
E
!±
(k) = cos
µ
Z
k⌧1
1.Vacuum birefringence
2.Vacuum dispersion
3.Fermionic saturation
ky
m≤1: mass
n-1: refraction index
kz
E
2~n k
2
2~n k
k
2
k
B
~vg (k)
kx
The LTM standards of the theory
Dimensionless variables
x[m]
x=
∈ Z,
a
t[sec]
∈ N,
t=
t
Relativistic limit:
Measure
from mass-refraction-index
n(m[kg] ) =
Measure
!
1−
"m
[kg]
m
#
from light-refraction-index
"
!
k
∓
c (k) = c 1 ± √
3kmax
m[kg]
m=
∈ [0, 1]
m
The relativity principle
Bisio, D'Ariano, Perinotti, arXiv:1503.01017
Symmetries and Relativity Principle
Looking for changes of reference-frames that leaves the dynamics invariant
Change of reference-frame = special change of representation
VA QWs
A=
Z
dk Ak
i
†
n(k) · T := (Ak − Ak ) “Hamiltonian”
2
n(k) analytic in k
µ
B
(I, T) = (T )
Dynamics: eigenvalue equation
iω
Ak ψ(k, ω) = e ψ(k, ω)
Hermitian basis for
Lin(C )
For each value of k there are
at most s eigenvalues {ωl (k)}
n(k) analytic in k + finite-dim irreps.
ωl (k)
(sin ωI − n(k) · T)ψ(k, ω) = 0
s
continuous
dispersion relations branches
Bisio, D'Ariano, Perinotti, arXiv:1503.01017
Symmetries and Relativity Principle
Change of reference-frame:
{Lβ }β∈G
(ω, k) → (ω ′ , k′ ) = Lβ (ω, k)
Lβ invertible (generally non continuous) over [−π, π] × B
G
group (including space-inversion, charge conjugation,…)
Symmetry of the dynamics:
there exists a pair of invertible matrices
such that the following identity holds:
(sin ωI − n(k) · T) =
Γβ and Γ̃β
G0
(id-component of
can also contain
LUs, gauge-transforms, …,
Γβ and Γ̃β
−1
′
Γ̃β (sin ω I
continuous functions of
Γ̃β , Γβ
(ω, k)
− n(k ) · T)Γβ
G ) preserves the branches
′
change of reference-frame = k → k (k)
′
change of reference-frame = reshuffling k → k′ (k) of
irreps. holds for the whole class of VA QW
′
k → k (k)
Lβ (ω, k) = (ω(k′ ), k′ (k))
G0 , G
depend on
the QW!
Bisio, D'Ariano, Perinotti, arXiv:1503.01017
Relativity Principle for Weyl QW
p
(f )
µ
pµ p = 0
:= f (ω, k)(sin ω, n(k))
on Disp(A)
) µ
p(f
µ σ ψ(k, ω) = 0 “4-momentum”
Non-linear Lorentz group
(f )
Lβ
:= D
(f )−1
Lβ D
D
(f )
(f )
: (ω, k) !→ p
acting on [−π, π] × B
Disp(A) invariant
(f )
(ω, k)
Lβ
Lorentz
Relativistic covariance of dynamics
(sin ωI − n(k) · σ) =
Λβ ∈ SL2 (C)
†
′
Λ̃β (sin ω I
independent of
(kµ )
′
− n(k ) · σ)Λβ
Bisio, D'Ariano, Perinotti, arXiv:1503.01017
Relativity Principle for Weyl QW
Includes the group of “translations” of the Cayley graph:
⇡
ky
0
⇡
kz
⇡
0
ky
0
⇡
⇡
kz
⇡
The Brillouin zone
separates into four
invariant regions
diffeomorphic to balls,
corresponding to four
different particles.
0
⇡
⇡
⇡
ky
0
⇡
⇡
0
kx
⇡
⇡
0
ky
⇡
0
⇡
⇡
G0 is the Poincaré group
⇡
2
kx
kz
⇡
⇡
⇡
2
kz
0
kz
0
ky 0
⇡
⇡
⇡
0
kx
⇡
⇡
2
0
⇡
kx
⇡
⇡
2
0
k
⇡
2
0
⇡
2
Bisio, D'Ariano, Perinotti, arXiv:1503.01017
Relativity Principle for Dirac QW
Dirac automaton: De Sitter covariance (non linear)
Covariance for Dirac QCA cannot leave m invariant
invariance of de Sitter norm:
Disp(A) :
invariance
for
Paolo Perinotti
Marco Erba
Alessandro Bisio
Franco Manessi
Alessandro Tosini
Nicola Mosco
D'Ariano and Perinotti, Derivation of the Dirac Equation from Principles of Information processing, Phys. Rev. A 90 062106 (2014)
Bisio, D'Ariano,Tosini, Quantum Field as a Quantum Cellular Automaton: the Dirac free evolution in 1d, Annals of Physics 354 244 (2015)
D’Ariano, Mosco, Perinotti, Tosini, Path-integral solution of the one-dimensional Dirac quantum cellular automaton, PLA 378 3165 (2014)
D’Ariano, Mosco, Perinotti, Tosini, Discrete Feynman propagator for the Weyl quantum walk in 2 + 1 dimensions, EPL 109 40012 (2015)
D'Ariano, Manessi, Perinotti, Tosini, The Feynman problem and Fermionic entanglement …, Int. J. Mod. Phys. A17 1430025 (2014)
Bibeau-Delisle, Bisio, D’Ariano, Perinotti, Tosini, Doubly-Special Relativity from Quantum Cellular Automata, EPL 109 50003 (2015)
Bisio, D'Ariano, Perinotti, Quantum Cellular Automaton Theory of Light, arXiv:1407.6928
Bisio, D'Ariano, Perinotti, Lorentz symmetry for 3d Quantum Cellular Automata, arXiv:1503.01017
D'Ariano, A Quantum Digital Universe, Il Nuovo Saggiatore 28 13 (2012)
D'Ariano, The Quantum Field as a Quantum Computer, Phys. Lett. A 376 697 (2012)
D'Ariano, Physics as Information Processing, AIP CP1327 7 (2011)
D’Ariano, On the "principle of the quantumness", the quantumness of Relativity, and the computational grand-unification, in AIP CP1232 (2010)