Download Semi-classical gravity based on deBroglie

Semi-classical gravity based on de Broglie-Bohm theory
Ward Struyve
Institute for theoretical physics
University of Leuven, Belgium
Non-relativistic de Broglie-Bohm theory
(a.k.a. pilot-wave theory, Bohmian mechanics, . . . )
• De Broglie (1927), Bohm (1952)
• Particles moving under influence of the wave function.
• Dynamics:
i~∂tψ =
−
n
X
k=1
~2 2
∇k + V
2mk
!
ψ
dxk
= vkψ (x1, . . . , xn, t),
dt
where
vkψ =
~
∇k ψ
1
Im
=
∇k S,
mk
ψ
mk
ψ = |ψ|eiS/~
• Double Slit experiment:
• Quantum equilibrium:
- for an ensemble of systems with wave function ψ
- distribution of particle positions ρ(x) = |ψ(x)|2
• Quantum equilibrium is preserved by the particle motion (= equivariance), i.e.
ρ(x, t0) = |ψ(x, t0)|2
⇒
ρ(x, t) = |ψ(x, t)|2
∀t
• Agreement with quantum theory in quantum equilibrium. Deviations are possible in non-equilibrium.
• Classical limit:
ẋ =
ψ = |ψ|e
1
∇S
m
iS/~
,
⇒
mẍ = −∇(V + Q)
~2 ∇2|ψ|
Q=−
= quantum potential
2m |ψ|
Classical trajectories when |∇Q| |∇V |.
• Wave function of subsystem: conditional wave function
Consider composite system: ψ(x1, x2, t), (X1(t), X2(t))
Conditional wave function for system 1:
χ(x1, t) = ψ(x1, X2(t), t)
The trajectory X1(t) satisfies
dX1(t)
1
∇
χ(x
,
t)
1
1
= v χ(X1(t), t) =
Im
dt
m1
χ(x1, t) Conditional wave function χ:
- will satisfy Schrödinger equation in certain cases
- will undergo collapse
x1 =X1 (t)
Relativistic de Broglie-Bohm theory: Dirac particles
• Single particle:
Dirac equation:
iγ µ∂µψ − mψ = 0
Guidance equation:
dxµ
∼ j µ = ψ †γ 0γ µψ
dτ
→ Is fully Lorentz invariant!
Relativistic de Broglie-Bohm theory: Dirac particles
• Single particle:
Dirac equation:
i∂µγ µψ − mψ = 0
Guidance equation:
dxµ
∼ j µ = ψ †γ 0γ µψ
dτ
→ Is fully Lorentz invariant!
• Many particles:
→ Difficult to have fundamental Lorentz invariance. (Due to quantum non-locality.)
Guidance equation is formulated with respect to preferred reference frame.
But in quantum equilibrium the statistical predictions are Lorentz invariant
(preferred frame can not be detected).
Can the theory made Lorentz invariant?
(Dürr, Goldstein, Norsen, Struyve & Zanghı̀, in prep.)
• Quantum equilibrium can not hold in all reference frames. Berndl et al. (1996)
• Dynamics of the particles can be choosen such that equilibrium holds with respect to a particular reference frame or foliation F (determined by normal nµ).
Ẋk (Σ) = vkψ,F (X1(Σ), . . . , XN (Σ))
→ velocity v ψ,F depends on wave function and foliation
• Lorentz invariant law for foliation F. E.g.
1 The foliation is a hyperplane foliation (∂ν nµ = 0).
2 The foliation is arbitrary.
→ Not very serious.
• Lorentz invariant law for foliation F. E.g.
1 The foliation is a hyperplane foliation (∂ν nµ = 0).
2 The foliation is arbitrary.
→ Not very serious.
• Instead of foliation as extra structure, let foliation be completely
determined by the wave function, i.e. F = F(ψ)
Covariant map ψ → F = F(ψ):
ψ
 −→ F(ψ)



U (Λ)y
yΛ
ψ 0 −→ F(ψ 0)
For example:
nµ ∼ P µ, P µ the total expected energy momentum 4-vector,
Z
b
P µ = hψ|H|ψi,
hψ|b
p|ψi = hψ| dσν Tbνµ(x)|ψi
σ
Semi-classical gravity
There are conceptual difficulties with the quantum treatment of gravity: problem
of time, finding solutions to Wheeler-DeWitt equation, etc. Therefore one often
resorts to semi-classical approximations:
→ Matter is treated quantum mechanically, as quantum field on curved
space-time.
E.g. scalar field:
b g)Ψ(φ, t)
i∂tΨ(φ, t) = H(φ,
→ Grativity is treated classically, described by
8πG
Gµν (g) = 4 hΨ|Tbµν (φ, g)|Ψi
c
Is there a better (i.e. more refined) semi-classical approximation
based on de Broglie-Bohm theory?
In de Broglie-Bohm theory matter is described by Ψ(φ) and actual scalar field
φB (x, t). Proposal for semi-classical theory:
8πG
Gµν (g) = 4 Tµν (φB , g)
c
Is there a better (i.e. more refined) semi-classical approximation
based on de Broglie-Bohm theory?
In de Broglie-Bohm theory matter is described by Ψ(φ) and actual scalar field
φB (x, t). Proposal for semi-classical theory:
8πG
Gµν = 4 Tµν (φB )
c
→ In general doesn’t work because ∇µT µν (φB ) 6= 0!
(In non-relativistic de Broglie-Bohm theory energy is not conserved.)
Is there a better (i.e. more refined) semi-classical approximation
based on de Broglie-Bohm theory?
In de Broglie-Bohm theory matter is described by Ψ(φ) and actual scalar field
φB (x, t). Proposal for semi-classical theory:
8πG
Gµν = 4 Tµν (φB )
c
→ In general doesn’t work because ∇µT µν (φB ) 6= 0!
(In non-relativistic de Broglie-Bohm theory energy is not conserved.)
Similar situation in scalar electrodynamics:
Quantum matter field described by Ψ(φ) and actual scalar field φB (x, t). Semiclassical theory:
∂µF µν = j ν (φB )
→ In general doesn’t work because ∂ν j ν (φB ) 6= 0!
Outline
• Semi-classical approximation in non-relativistic de Broglie-Bohm
• Semi-classical approximation in scalar electrodynamics
• Semi-classical approximation in mini-superspace model
Semi-classical approximation: non-relativistic quantum mechanics
• System 1: quantum mechanical. System 2: classical
Usual approach (mean field):
2
∇1
+ V (x1, X2(t)) ψ(x1, t)
i∂tψ(x1, t) = −
2m1
m2Ẍ2(t) = hψ|F2(x1, X2(t))|ψi ,
F2 = −∇2V
→ backreaction through mean force
Semi-classical approximation: non-relativistic quantum mechanics
• System 1: quantum mechanical. System 2: classical
Usual approach (mean field):
2
∇
i∂tψ(x1, t) = − 1 + V (x1, X2(t)) ψ(x1, t)
2m1
m2Ẍ2(t) = hψ|F2(x1, X2(t))|ψi ,
F2 = −∇2V
→ backreaction through mean force
de Broglie-Bohm-based approach:
2
∇
i∂tψ(x1, t) = − 1 + V (x1, X2(t)) ψ(x1, t)
2m1
Ẋ1(t) = v1ψ (X1(t), t) ,
m2Ẍ2(t) = F2(X1(t), X2(t))
→ backreaction through Bohmian particle
• Prezhdo and Brookby (2001):
De Broglie-Bohm-based approach yields better results than usual approach:
• Derivation of de Broglie-Bohm-based semi-classical approximation
Full quantum mechanical description:
2
2
∇2
∇1
−
+ V (x1, x2) ψ(x1, x2, t)
i∂tψ(x1, x2, t) = −
2m1 2m2
Ẋ1(t) = v1ψ (X1(t), X2(t), t) ,
Ẋ2(t) = v2ψ (X1(t), X2(t), t)
Conditional wave function χ(x1, t) = ψ(x1, X2(t), t) satisfies
2
∇
i∂tχ(x1, t) = − 1 + V (x1, X2(t)) χ(x1, t) + I(x1, t)
2m1
and particle two:
m2Ẍ2(t) = −∇2V (X1(t), x2)
−∇2Q(X1(t), x2)
x2 =X2 (t)
x2 =X2 (t)
→ Semi-classical approximation follows when I and −∇2Q are negligible
(e.g. when particle 2 is much heavier than particle 1)
Semi-classical approximation: scalar electrodynamics
• Classical field equations:
DµDµφ + m2φ = 0 ,
∂µF µν = j ν = ie (φ∗Dν φ − φDν∗φ∗)
with Dµ = ∂µ + ieAµ, F µν = ∂ µAν − ∂ ν Aµ and j µ charge current (∂µj µ = 0).
• Quantum field theory in Coulomb gauge ∇ · A = 0:
...
• De Broglie-Bohm approach:
– Wave functional: Ψ(φ, AT , t)
– Actual field configurations: φ(x), AT (x)
...
• De Broglie-Bohm-based semi-classical approximation:
– Matter field: quantum mechanical; Electromagnetic field: classical
– Schrödinger equation for Ψ(φ, t):
Z
2
δ
1 1
3
T
2
2
2
i∂tΨ = d x − ∗ + |(∇ − ieA )φ| + m |φ| − C 2 C Ψ ,
δφ δφ
2 ∇
where C is the charge density operator.
– Guidance equation scalar field:
δS
1
φ̇ = ∗ − eφ 2 CS
δφ
∇
⇒
DµDµφ − m2φ = −
δQ
δφ∗
– Classical Maxwell equations with quantum correction:
∂µF µν = j ν +jQν ,
with “quantum charge current” jQν = (0, jQ):
Is consistent since: ∂µ(j µ + jQµ ) = 0.
jQ = i∇ ∇12 CQ
• De Broglie-Bohm-based semi-classical approximation:
– Matter field: quantum mechanical; Electromagnetic field: classical
– Schrödinger equation for Ψ(φ, t):
Z
2
1 1
δ
i∂tΨ = d3x − ∗ + |(∇ − ieAT )φ|2 + m2|φ|2 − C 2 C Ψ ,
δφ δφ
2 ∇
where C is the charge density operator.
– Guidance equation scalar field:
δS
1
δQ
µ
2
φ̇ = ∗ − eφ 2 CS
⇒
DµD φ − m φ = − ∗
δφ
∇
δφ
– Classical Maxwell equations with quantum correction:
∂µF µν = j ν +jQν ,
with “quantum charge current” jQν = (0, jQ):
jQ = i∇ ∇12 CQ
Is consistent since: ∂µ(j µ + jQµ ) = 0.
→ Crucial in the derivation was that gauge was eliminated!
How to eliminate it in canonical quantum gravity?
(in that case: gauge = spatial diffeomorphism invariance).
Semi-classical approximation: mini-superspace model
• Restriction to homogeneous and isotropic (FRW) metrics and fields:
– Gravity: ds2 = −N (t)2dt2 + a(t)2dΩ23
– Matter: φ = φ(t)
Wheeler-DeWitt equation:
(HG + HM )ψ = 0 ,
1
3
∂
(a∂
)
+
a
VG ,
a
a
2
4a
Guidance equations (N = 1):
1
ȧ = − ∂aS ,
2a
• Semi-classical approximation:
HG =
HM = −
φ̇ =
1 2
3
∂
+
a
VM
φ
3
2a
1
∂φS
3
a
1
i∂tψ = HM ψ ,
φ̇ = 3 ∂φS
a
and Friedmann equation with quantum correction:
ȧ2 φ̇2
=
+ VM + VG+Q
a2
2
Summary
Results so far:
• Consistent semi-classical approximation for:
– Non-relativistic systems
– Quantum electrodynamics
– Mini-superspace models
To do:
• Find semi-classical approximation for full quantum gravity
• Find higher order correction terms
• Develop rigorous expansions (e.g. à la WKB)
• Find applications (e.g. in cosmology: Hawking radiation, cosmological perturbations in inflation theory)