On The Conceptual Issues Surrounding the Notion of

.
.
On The Conceptual Issues Surrounding the Notion of
Background Independent Bohmian Dynamics
Antonio Vassallo [Joint work with Pui Him Ip]
[email protected]
Section de Philosophie
Université de Lausanne
June, 26th 2014
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
1 / 20
Setting the stage
Natural philosophy: An approach to the construction and
interpretation of physical theories that treats physics and metaphysics
as a seamless whole ⇒ Ontological clarity.
Primitive ontology: A method of constructing physical theories
involving a dual structure (X , E).
X is the primitive ontology properly said;
E is a law-like object that “guides” (some of) the elements of X .
What does the term “primitive ontology” exactly mean?
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Setting the stage
The threefold meaning of “primitive ontology”:
1. The privileged variables in the formalism.
2. What is “guided” by E.
. What there is.
3
.
Desideratum
.
These
three meanings have to be compatible.
.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Bohmian Mechanics
Bohmian Mechanics (BM) is the simplest Galilean-invariant theory of
moving point-like particles.
The dynamics fixes the particles’ trajectories in (neo-)Newtonian
spacetime as the integral curves of the velocity field generated by a
wave function Ψ, solution of the Schrödinger equation.
The theory describes the evolution of a universal configuration of
particles.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Bohmian Mechanics
Let us try to construct BM using the canonical quantization procedure.
Lagrangian: L = T − V .
Configuration variables: {q1 , . . . , qN , t} ∈ R3N × R.
Parametrized Lagrangian: L′ =
Conjugate momenta: pi =
∂L′
dq
∂ d λi
dt
dλ L
(λ ∈ R).
(i = 1, . . . , N, t).
Hamiltonian constraint: X = H − pt ≈ 0.
Consider Ψ ∈ L2 (R3N × R, µ).
Turn canonical variables into operators:
qi → qˆi Ψ = qi Ψ;
pi → pˆi Ψ = iℏ
∂Ψ
;
∂qi
pˆt Ψ = iℏ
∂Ψ
.
∂t
ˆ − iℏ ∂ )Ψ = 0.
ˆ = (H
Implement the constraint: X ≈ 0 → XΨ
∂t
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Bohmian Mechanics
That’s not the entire story! We still have to provide a law for the
evolution of the primitive ontology: Q ≡ {q1 , . . . , qN } ∈ R3N .
Hence, we construct the simplest velocity field
on Ψ such that it is invariant under:
dQ(t)
dt
= vΨ depending
. Translations and rotations in Euclidean 3-space.
. Temporal reflections.
3. Galilean boosts.
1
2
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Bohmian Mechanics
.
Dynamical Equations
.
∂Ψ(Q(t), t)
ˆ
= HΨ(Q(t),
t),
∂t
d Q(t)
ℏ ∇Ψ(Q(t), t)
= Im
.
dt
m
Ψ(Q(t), t)
iℏ
.
Q ≡ {q1 , . . . , qN } ∈ Q = R3N .
m ≡ {m1 , . . . , mN }.
∇ ≡ { ∂q∂ 1 , . . . , ∂q∂N }.
Ψ(Q(t), t) : R3N × R → C.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Bohmian Mechanics
. The privileged variables in the formalism: Positions in Euclidean
3-space.
2. What is “guided” by Ψ: Point-like particles.
3. What there is: Point-like particles plus an absolute Newtonian
1
background.
There is a tension between 2 and 3.
Absolute backgrounds create metaphysical discomfort.
Spatiotemporal backgrounds are untenable when moving to the
(canonical) quantum gravitational regime.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Bohmian Mechanics
Can we construct a version of the theory that dispenses with the
Newtonian background? Yes.
Two strategies:
Eliminate space and time from primitive ontology (relational Bohmian
Mechanics).
Put spatiotemporal degrees of freedom among the “guided” stuﬀ
(Bohmian Canonical Quantum Gravity).
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Relational Particle Mechanics
Step 1: Elimination of the preferred embedding in Euclidean 3-space.
Particles are individuated with respect to their relative distances (a
configuration is individuated independently of its embedding in
Euclidean 3-space).
Hence, the configuration space of the theory Q0 has to be R3N
quotiented by the group of (at least) translations and rotations in
Euclidean space.
A curve in Q0 is a sequence of universal configurations of particles
given in terms of interparticle separations.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
10 / 20
Relational Particle Mechanics
Step 2: Elimination of the privileged temporal metric.
If we admit an external universal time, then the same curve in Q will
correspond to several curves in Q × R (same configurational evolution
at diﬀerent time rates).
If we want to eschew absolute external time, we must claim that the
“real” dynamics happens in Q0 , and, hence, it is timeless.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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Relational Particle Mechanics
Antonio Vassallo (UNIL)
Background Independent BM
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Relational Particle Mechanics
Timeless dynamics in Q0 : stacking two configurations on top of one
another in a way that extremizes the absolute diﬀerence.
To determine the diﬀerence, use “best matching procedure”:
. “Intrinsic” diﬀerence from all the possible variations:
v
u N
u1 ∑
t
mi δqi · δqi .
2
1
i=1
. Extremum “intrinsic” diﬀerence as the best matching:
v
∫ √ u
N
u1 ∑
∑
∑
εα Oα qi ) = 0.
εα Oα qi ) · (δqi −
mi (δqi −
δI = δ
FEt
2
α
α
2
i=1
With FE = E − V (Oα being the generators of the Euclidean group).
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
13 / 20
Relational Particle Mechanics
This is equivalent to extremizing the following family of actions on Q:
∫ √
S=
FE T d λ;
1 ∑ d qi d qi
T =
mi
.
2
dλ dλ
N
i=1
It can be shown that, if we choose a parameter such that
√
∑N
i=1 mi δqi · δqi
δλ =
,
2FE
we recover Newton’s equations of motion.
To recover change in time means to make an unique choice of λ.
“Time” is a derived quantity in the perspective of timeless mechanics.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
14 / 20
Relational Bohmian Mechanics
Let us try to canonically quantize the theory.
√
1. Lagrangian: L =
FE T .
i
√ mi dq
∂L )
(
2. The momenta are then p =
=
FE √Td λ .
i
dqi
∂
dλ
3. Hamiltonian constraint: T − F
E = T − E + V = H − E ≈ 0.
ˆ = E Ψ.
4. Implementation of the constraint: H − E ≈ 0 → HΨ
. If the total energy of the system is zero, then we obtain an equation
that resembles the Wheeler-DeWitt equation.
5
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
15 / 20
Relational Bohmian Mechanics
What about the guiding equation?
Nota bene: Ψ is defined over Q.
In order to find a wave function defined over Q0 , we need to
implement a best matching procedure that defines and extremizes the
“intrinsic” diﬀerence between Ψ(Q) and Ψ(Q + δQ).
But we also have primitive stuﬀ, namely, universal instantaneous
relational configurations represented by points in Q0 .
The wave function does not determine any velocity for the particles, it
just detemines the “piling” of universal configurations according to a
dynamically selected curve in Q0 .
These facts could be implemented by a procedure that selects a vector
field in Q and then best matches its integral curves to a single curve
in Q0 .
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
16 / 20
Relational Bohmian Mechanics
. The privileged variables in the formalism: Interparticle separations.
2. What is “guided” by Ψ: Instantaneous universal configurations.
3. What there is: Point particles plus separation relation.
1
RBM’s dynamics implements non-locality in a Bohmian sense: the
summation in the best-matching action principle means that all the
particles in the universe as a whole contribute to the determination of
the timeless history.
BM and RBM have radically diﬀerent primitive ontologies, whatever
the sense we pick out.
Unlike BM, there is no tension between 1,2, and 3.
The relational approach to BM delivers encouraging results towards
the implementation of a Bohmian theory of quantum gravity.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
17 / 20
Bohmian Quantum Geometrodynamics
.
Dynamical equations
.
ˆ
HΨ[h]
= 0,
.
dh
δΨ[h]
= N × DeW[h] × ImΨ[h]−1
.
dτ
δh
h ∈ Q = Riem(Σ3 )/diﬀ (Σ3 ).
DeW[h] ⇒ Gabcd = (hac hbd + had hbc − hab hcd ).
N is the so-called lapse function: It specifies a stacking of
3-geometries.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
18 / 20
Bohmian Quantum Geometrodynamics
. The privileged variables in the formalism: Riemannian 3-metrics.
2. What is “guided” by Ψ: Riemannian 3-geometries.
3. What there is: Riemannian 3-geometries plus a time-like flow.
1
The theory might potentially include matter
by expanding the
⊕
configuration space to Q = Riem(Σ3 ) Q0 .
However, the theory postulates more spatiotemporal structure than a
genuinely general relativistic one.
Can a relational approach mitigate this problem?
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
19 / 20
References
Barbour, J. (2012). Shape dynamics. an introduction. In F. Finster,
O. Müller, M. Nardmann, J. Tolksdorf, and E. Zeidler (Eds.), Quantum
field theory and gravity, pp. 257–297. Birkhäuser.
http://arxiv.org/abs/1105.0183.
Barbour, J. and B. Bertotti (1982). Mach’s principle and the structure of
dynamical theories. Proceedings of the royal society A 382, 295–306.
Dürr, D., S. Goldstein, and N. Zanghì (2013). Quantum physics without
quantum philosophy. Springer.
Goldstein, S. and S. Teufel (2001). Quantum spacetime without observers:
ontological clarity and the conceptual foundations of quantum gravity. In
C. Callender and N. Huggett (Eds.), Physics meets Philosophy at the
Planck scale, pp. 275–289. Cambridge University Press.
http://arxiv.org/abs/quant-ph/9902018. Reprinted in Dürr et al.
(2013), chapter 11.
Antonio Vassallo (UNIL)
Background Independent BM
June, 26th 2014
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