Quantum Field Theories as Statistical Field Theories · Quantum Field Theories as Statistical Field...

Quantum Field Theoriesas

Statistical Field Theories

Antoine TilloyMax Planck Institute of Quantum Optics, Germany

Oberseminar Mathematische Physik LMU, Munich, GermanyFebruary 1, 2017

First problem:

Quantum field theory is not about fields, there are no fields, it’sabout correlation functions of ... macroscopic stuff made of ...

⟨ϕ(x1)ϕ(x2)ϕ(x3)ϕ(x4)⟩ =∑

· · ·

⟨ψ(x1)ψ(x2)ψ(x3)ψ(x4)⟩ =∑

· · ·

what we want what we get

First problem:

Quantum field theory is not about fields, there are no fields, it’sabout correlation functions of ... macroscopic stuff made of ...

⟨ϕ(x1)ϕ(x2)ϕ(x3)ϕ(x4)⟩ =∑

· · ·

⟨ψ(x1)ψ(x2)ψ(x3)ψ(x4)⟩ =∑

· · ·

what we want what we get

Second problem:

Even as operational theories, “physical” interacting quantum fieldtheories are ill defined.

⟨ϕ(x1)ϕ(x2)ϕ(x3)ϕ(x4)⟩ =+∞∑k=0

(∞)k gk

⟨ψ(x1)ψ(x2)ψ(x3)ψ(x4)⟩ =+∞∑k=0

(∞)k gk

These are actually the two classes of difficulties noted by Paul Dirac in 1963

Second problem:

Even as operational theories, “physical” interacting quantum fieldtheories are ill defined.

⟨ϕ(x1)ϕ(x2)ϕ(x3)ϕ(x4)⟩ =+∞∑k=0

(∞)k gk

⟨ψ(x1)ψ(x2)ψ(x3)ψ(x4)⟩ =+∞∑k=0

(∞)k gk

These are actually the two classes of difficulties noted by Paul Dirac in 1963

Reformulations of quantum theory are needed

The Bohmian way

dQk(t)dt = vΨk (Q1(t), · · · ,Qn(t))

iℏ dΨdt = HΨ

L. de Broglie 1927

D. Bohm 1952

The Bohmian way

dQk(t)dt = vΨk (Q1(t), · · · ,Qn(t))

iℏ dΨdt = HΨ

L. de Broglie 1927

D. Bohm 1952

Subtleties

Lorentz invariance

Either fixed foliation of foliation determined by the wave function →the letter but not the spirit

Particle creation-annihilation 2 solutions

∙ Stochastic creation-annihilation events∙ Resurrect the Dirac sea

Subtleties

Lorentz invariance

Subtleties

Lorentz invariance

Particle creation-annihilation

2 solutions

Subtleties

Lorentz invariance

The Collapse way

A modified Schrödinger equation

∂t |ψw⟩ = −iH |ψw⟩+ tiny(w, ψw)

A primitive ontology

fields, particles, flashes

The Collapse way

What is known:

A Lorentz invariant GRW (Tumulka, 2006)

but no interactions and a wave function formalism

A Lorentz invariant CSL (Bedingham, 2011)

but a non-linearity in the smearing function making the statisticalinterpretation unclear

In both cases anyway, predictions = QFT.

What is known:

Objective:

Construct a relativistic collapse model, naturally written as aLorentz invariant statistical field theory, that can be embeddedwithin an “orthodox” interacting quantum field theory.

In provocative form

1. Collapse models can be made equivalent to quantum theory2. Quantum field theories can be written as statistical field

theories3. The two previous points are equivalent

Objective:

In provocative form

Objective:

In provocative form

1. Collapse models can be made equivalent to quantum theory

2. Quantum field theories can be written as statistical fieldtheories

3. The two previous points are equivalent

Objective:

In provocative form

theories

3. The two previous points are equivalent

Objective:

In provocative form

dynamical reduction models

an instantiation: the csl model

Linear collapse equation:

ddt |ψw(t)⟩ =

−iH0 +

d3x Mσ(x)wt(x)white

−√λ

2 M2σ(x)

|ψw(t)⟩ ,

with Mσ(x) =1

σ(2π)32

N∑k=1

dyk mk e−|x−yk|

2σ2 |yk⟩ ⟨yk| ,

Master equation for ρt = E[|ψw(t)⟩ ⟨ψw(t)|

]ddt ρ(t) = −i [H0, ρ(t)]−

d3x[Mσ(x), [Mσ(x), ρ(t)]

More generally ρt = Φt · ρ0with Φ linear Completely Positive Trace Preserving

needs to be an “unraveling” of a nice open-system evolution

Gisin Diosi

ddt |ψw(t)⟩ =

−iH0 +

−√λ

2 M2σ(x)

|ψw(t)⟩ ,

with Mσ(x) =1

σ(2π)32

N∑k=1

dyk mk e−|x−yk|

2σ2 |yk⟩ ⟨yk| ,

]ddt ρ(t) = −i [H0, ρ(t)]−

Gisin Diosi

ddt |ψw(t)⟩ =

−iH0 +

−√λ

2 M2σ(x)

|ψw(t)⟩ ,

with Mσ(x) =1

σ(2π)32

N∑k=1

dyk mk e−|x−yk|

2σ2 |yk⟩ ⟨yk| ,

]ddt ρ(t) = −i [H0, ρ(t)]−

Gisin Diosi

ddt |ψw(t)⟩ =

−iH0 +

d3x Mσ(x)wt(x)−√λ

2 M2σ(x)

|ψw(t)⟩ ,

Normalization and cooking:

∣∣∣ψw(t)⟩=

|ψw(t)⟩√⟨ψw(t)|ψw(t)⟩

dµt(w) = ⟨ψw(t)|ψw(t)⟩ · dµ0(w)

wt(x) = 2√λ ⟨Mσ(x)⟩+ bt(x)

white.

ddt |ψw(t)⟩ =

−iH0 +

d3x Mσ(x)wt(x)−√λ

2 M2σ(x)

|ψw(t)⟩ ,

Normalization and cooking:

∣∣∣ψw(t)⟩=

|ψw(t)⟩√⟨ψw(t)|ψw(t)⟩

dµt(w) = ⟨ψw(t)|ψw(t)⟩ · dµ0(w)

wt(x) = 2√λ ⟨Mσ(x)⟩+ bt(x)

white.

The “good” choice of primitive ontology is w, not ⟨M(x)⟩:

∙ Is the natural object appearing in the linear eq.∙ Does projects down to R3 the localization of the state:

wt(x) = 2√λ ⟨Mσ(x)⟩+ bt(x)

∙ Allows to reconstruct the state exactly: w → |ψw⟩∙ Allows a time symmetric formulation of collapse

Bedingham & Maroney (2015)

∙ Is the natural Lorentz invariant object in generalizationsTumulka (2006), Bedingham (2011)

∙ Allows quantum classical couplingAT & Diósi (2016)

∙ Is the natural object appearing in the linear eq.

∙ Does projects down to R3 the localization of the state:wt(x) = 2

√λ ⟨Mσ(x)⟩+ bt(x)

wt(x) = 2√λ ⟨Mσ(x)⟩+ bt(x)

∙ Allows to reconstruct the state exactly: w → |ψw⟩

∙ Allows a time symmetric formulation of collapseBedingham & Maroney (2015)

wt(x) = 2√λ ⟨Mσ(x)⟩+ bt(x)

a recipe for collapse models

Start with a quantum system

Couple it with a bosonic bath

Trace out the bath

Stochastically unravel the evolution

Normalize the state and cook the measure

Keep only the primitive ontology

→ you now have a single cat, living in R3

Keep only the primitive ontology

→ you now have a single cat, living in R3

changing the recipe for qft

Take an interacting QFT:

Split fermions and bosons:

→ now do the same as before with the bosons as bath

Split fermions and bosons:

→ now do the same as before with the bosons as bath

Final picture:

This time without changing the empirical content of the theory!

Final picture:

This time without changing the empirical content of the theory!

in practice

Consider a Yukawa theory:

Lf(ψ, ∂µψ) = ψ(i/∂ − mf)ψ

Lint(ψ, ϕ) = g ψ ψ ϕ

Lb(ϕ, ∂µϕ) =12∂µϕ∂

µϕ− 12m2

Take the momentum regularization Λ as fundamental

in practice

Consider a Yukawa theory:

Lf(ψ, ∂µψ) = ψ(i/∂ − mf)ψ

Lint(ψ, ϕ) = g ψ ψ ϕ

Lb(ϕ, ∂µϕ) =12∂µϕ∂

µϕ− 12m2

Take the momentum regularization Λ as fundamental

in practice

1) Integrating out the bosons

Define ρt = trb[ρtot].

Wick’s theorem gives:

ρf(t) = T exp(iΦ

[jL, jR

])· ρf(ti)

where Φ[jL, jR] is the “operator” influence phase functional:

iΦ[jL, jR

d4x d4y D(x, y) jL(x)jR(y)

− 12θ(x

0 − y0)D(x, y) jL(x)jL(y)− 12θ(y

0 − x0)D(x, y) jR(x)jR(y)

with D(x, y) = trb

[ϕ(x)ϕ(y) ρb(ti)

in practice

Define ρt = trb[ρtot]. Wick’s theorem gives:

ρf(t) = T exp(iΦ

[jL, jR

])· ρf(ti)

iΦ[jL, jR

− 12θ(x

0 − y0)D(x, y) jL(x)jL(y)− 12θ(y

with D(x, y) = trb

[ϕ(x)ϕ(y) ρb(ti)

in practice

Define ρt = trb[ρtot]. Wick’s theorem gives:

ρf(t) = T exp(iΦ

[jL, jR

])· ρf(ti)

iΦ[jL, jR

− 12θ(x

0 − y0)D(x, y) jL(x)jL(y)− 12θ(y

with D(x, y) = trb

[ϕ(x)ϕ(y) ρb(ti)

in practice

2) Unraveling the open evolution

|ψξ(t)⟩ =T exp− i

d4x ȷ(x) ξ(x)

−∫ t

d4x d4y θ(x0 − y0) [D − S](x, y) ȷ(x)ȷ(y)|ψξ(t)⟩

E[ξ(x)ξ∗(y)] = D(x, y),E[ ξ(x) ξ(y) ] = S(x, y).

Note: importantly, finding ξ with the correlation matrix D is alwayspossible (the relation matrix S is a free parameter)

in practice

2) Unraveling the open evolution

|ψξ(t)⟩ =T exp− i

d4x ȷ(x) ξ(x)

−∫ t

d4x d4y θ(x0 − y0) [D − S](x, y) ȷ(x)ȷ(y)|ψξ(t)⟩

E[ξ(x)ξ∗(y)] = D(x, y),E[ ξ(x) ξ(y) ] = S(x, y).

Note: importantly, finding ξ with the correlation matrix D is alwayspossible (the relation matrix S is a free parameter)

in practice

3) Cook the measure

dµt(ξ) = ⟨ψξ(t)|ψξ(t)⟩dµti(ξ)

Finally push ti → −∞, t → +∞

In functional integral representation:

dµ+∞ ∝∣∣∣∣∫ D[ψ]D[ψ]κ[ψψ] exp

(iSf + i

∫d4x Jψ + Jψ − gψψξ

)∣∣∣∣2dµ−∞

in practice

3) Cook the measure

(iSf + i

)∣∣∣∣2dµ−∞

in practice

3) Cook the measure

(iSf + i

)∣∣∣∣2dµ−∞

in practice

One can prove collapse in position and amplification

Collapse “strength” Ω between two points x and y:

Ω+∞(x,y) = −g2

(2π)2

(K0 (mb|x − y|)− K0 (Λ|x − y|)− log

in practice

One can prove collapse in position and amplification

Collapse “strength” Ω between two points x and y:

Ω+∞(x,y) = −g2

(2π)2

(K0 (mb|x − y|)− K0 (Λ|x − y|)− log

discussion

reconsidering collapse

∙ Collapse can be seen as an interpretation of QFT∙ Expected empirical signatures of collapse computed so far areartifacts of Markovianity / frame dependence.

∙ Collapse can be seen as an interpretation of QFT

∙ Expected empirical signatures of collapse computed so far areartifacts of Markovianity / frame dependence.

∙ Collapse can be seen as an interpretation of QFT∙ Expected empirical signatures of collapse computed so far areartifacts of Markovianity / frame dependence.

reconsidering qft

1) QFTs can be seen as theories about fields

2) QFTs can be fundamentally UV-regularized

No Ostrogradsky instability, i.e. no need for:

→ makes Wilsonian renormalization transparent

reconsidering qft

limitations

∙ So far, harder to analyse with QED

∙ Not trivial to connect with physical parameters →renormalization

∙ It would be better to work from first principles

limitations

∙ So far, harder to analyse with QED∙ Not trivial to connect with physical parameters →renormalization

limitations

∙ So far, harder to analyse with QED∙ Not trivial to connect with physical parameters →renormalization

Summary

One can construct “simple” relativistic collapse models

They can be made equivalent to QFT

This gives an interpretation of QFT as a SFT

Still analytical, numerical and conceptual cleaning to do

Summary

Quantum Field Theories as Statistical Field Theories · Quantum Field Theories as Statistical Field...

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