Relativistic Quantum Physics

Relativistic quantum theory: particles or fields?Deeper troubles: non-locality and the MP

Relativistic Quantum Physics

Christian Wüthrich

Université de Genève

4th International Summer School in Philosophy of PhysicsSaig, 21 July 2016

Christian Wüthrich Relativistic Quantum Physics


My sources (without detailed credit)

Anthony Duncan (2012). The Conceptual Framework of Quantum Field Theory.OUP: Oxford.

Lawrence P Horwitz (2015). Relativistic Quantum Mechanics. Springer:Dordrecht.

Tom Lancaster and Stephen J Blundell (2014). Quantum Field Theory for theGifted Amateur. OUP: Oxford.

Lewis H Ryder (1985). Quantum Field Theory. CUP: Cambridge.

Steven Weinberg (1995). The Quantum Theory of Fields. Volume 1: Foundations.CUP: Cambridge.



John S Bell on what the real problem is

For me... this is the realproblem with quantum theory:the apparently essential con-flict between any sharp formu-lation and fundamental relativ-ity. That is to say, we have anapparent incompatibility, at thedeepest level, between the twofundamental pillars of contem-porary theory.

J S Bell. Speakable and unspeakable in quantummechanics. In his eponymous paper collection, CambridgeUP (2004): 172.



Troubles for the (single) particle interpretationFields

W B Yates

The innocent and the beautifulHave no enemy but time




Inspired by W B Yates

The particulate and the quantum mechanicalHave no enemy but spacetime




The superficial problem and its resolution

W G Unruh. Minkowski space-time and quantum mechanics. In V. Petkov (ed.), Minkowski Spacetime: AHundred Years Later, Springer (2010): 133-148.

In the setting of a quantum-mechanical theory of particles, theonly Lorentz-invariant interactions which conserve energy andmomentum are contact interactions between point particles(Unruh 2010, 137).

More generally, the tension between such a theory and SR canbe seen when comparing the wave function ψ(t , ~x) (where t is auniversal parameter, ~x are eigenvalues for position operator) andscalar field φ(t , ~x) (where t , ~x are coordinates of spt point).




Wave function vs. scalar fields

W G Unruh, “Minkowski space-time and quantum mechanics”, in V. Petkov (ed.), Minkowski Spacetime: AHundred Years Later, Springer (2010): 133-148.

Compare: wave function for two particles ψ(t , ~x , ~y) vs. pointinteractions of two (identical) scalar fields φ(t , ~x) · φ(t , ~x)

⇒ The single time t needed for the wave function of two particlesentail a need for a privileged foliation of spacetime; not so forfields.

Not an argument, just an indication. Nevertheless:

⇒ early recognition that a quantum field theory is needed to easethe tension with SR




The beginnings of relativistic quantum physics

1925: Schrödinger considers, but rejects, relativistic waveequation

1925: first quantum field theory (QFT) appears in the part writtenby Jordan in the ‘Drei-Männer-Arbeit’ with Born and Heisenberg

1926: Klein and Gordon (separately) propose their eponymousequation as description of relativistic electrons

1928: Dirac proposes his eponymous equation as description ofrelativistic electrons




Overheard at the 1927 Solvay conference:




Klein-Gordon equation for a single particle

E = p2/2m E2

c2 − p · p = m2c2

⇓ ⇓

E → i~ ∂∂t ,

p→ −i~∇.

⇓ ⇓

~2

2m∇2φ = −i~∂φ∂t (�−m2)φ = 0

(in natural units)




The changing fate of the probability interpretation

~2

2m∇2φ = −i~∂φ∂t (�−m2)φ = 0

(in natural units)

⇓ ⇓ρ = φ∗φ probability density ρ = i~

2m

(φ∗ ∂φ∂t − φ

∂φ∗

∂t

)⇓ ⇓

positive definite not positive definite

KG equation is second order, φ and ∂φ/∂t can be fixedarbitrarily at a time, and so ρ may become negative

⇒ interpretation of ρ as probability density must be given up

Also, ρ vanishes for a real φ; correct interpretation of ρ: chargedensity (complex φ corresponds to charged particle)




The Feynman-Stückelberger interpretation

Another problem with the KG equation is that the relativisticdispersion relation (regarded as equation for E), admits negativeenergy solutions:

E = ±(m2c4 + p2c2)1/2.

Feynman-Stückelberger: negative energy states as positiveenergy antiparticles, i.e., particles moving backward in time (withmomenta in opposite direction)

⇒ general solution of the KG equation is a superposition of twostates: incoming particle and outgoing antiparticle

⇒ general solution cannot be interpreted as description of singleparticle

better interpretation of KG equation: describes dynamics offields whose excitations are spinless particles




Further considerations against a single-particleinterpretation

TD Newton and E P Wigner. Localized states for elementary systems. Reviews of Modern Physics 21(1949):400-406.

Newton and Wigner add another reason that a relativisticsingle-particle QM is impossible:

for a zero-spin particle described by KG equation, there is asmall but non-vanishing probability of finding the particle whichwas originally located at the ‘origin’ outside its forward lightcone.




Further considerations against a single-particleinterpretation

Furthermore,

particle with mass m squeezed into a box smaller than itsCompton wavelength λ has a position uncertainty of ∆x � λ,and so ∆p � h/λ = m

Given this large energy (greater than its mass), SR suggeststhat particle/antiparticle pairs are being produced.

Thus, on average, the box contains more than one particle,challenging the single-particle interpretation.

⇒ All these considerations force abandoning a straightforwardparticle concept in relativistic quantum physics.




Guy Fawkes

A desperate disease requires a dangerous remedy.




Fields!

According to Lancaster and Blundell, QFT at core is the idea that“[e]very particle and every wave in the Universe is simply an excitationof a quantum field that is defined over all space and time.” (1)




What is a field?

A field is some kind of machinethat takes a position in space-time, given by the coordinate xµ,and outputs an object represent-ing the amplitude at that point inspacetime. Here the output is thescalar φ(xµ) but it could be, for ex-ample, a vector, a complex num-ber, a spinor or a tensor. [Quan-tum fields are captured by oper-ators which are locally defined,e.g., φ̂(xµ).]

(Caption in Lancaster and Blundell, 2)




Local relativistic quantum field theory

Local relativistic QFT is based on three principles (Duncan, 57f):

1 Quantum mechanics: linear superposition of amplitudes,probability interpretation of these amplitudes (squared), andunitary evolution of quantum state, implementing the dynamicsof the theory

2 Special relativity: symmetry of Lorentz invariance

3 ‘Clustering’: “insensitivity of local processes to the distantenvironment” (58); in context of relativistic theories, thisbecomes ‘locality’ or ‘microcausality’, which ensures, e.g.,absence of superluminal signalling




Consequences of the basic principles

forces resorting to approximative schemes (bad)

enforces constraints on Hamiltonian (good)

Two more important consequences:1 gives ‘explanation’ of existence of antimatter, with each

particle possessing an antiparticle of equal mass andopposite additive quantum numbers

2 spin-statistics theorem, clarifies contrasting symmetryproperties of the wavefunctions of particles of integer(bosonic) and half-integer (fermionic) spin




Antiparticles(From Duncan (2012,59f), from Weinberg 1972, going back to Feynman 1949)

Consider a process where a proton (P) emits a positive pion (π+)at spacetime event x , which then travels to y to be absorbed bya neutron (N):




‘locality’ here amounts to interaction between neutron andproton via local emission and absorption events of thirdintermediary particle

indeterminacy of position and velocity in QM permits for x and yto be space-like related




SR: Lorentz boost of left frame to frame on right changestemporal order of x and y so that y now precedes x

⇒ observer in right frame: negative pion (π−) emitted by neutron Nat y is later absorbed by proton P at x

⇒ “intimate association between spatiotemporal reflection andparticle-antiparticle interchange characteristic of local theoriesand exemplified in the [CPT] theorem” (60)




Locality and superluminality

But don’t the π’s travel superluminally and thus violate SR?



Locality and non-locality in QFTThe real threat: the measurement problem

At the heart of the tension: locality in axiomatic QFT

In axiomatic QFT, a ‘local action’ principle is usually observed,

where ‘local action’ means something like ‘no physical actioncan propagate faster than the speed of light’, or at least that‘there is no transmission of statistically measurable properties atfaster than light speed’, i.e., ‘no signalling’.

⇒ local actions in one spacetime region cannot changemeasurable properties in regions spacelike to it

⇒ operators in spacelike related regions commute




At the heart of the tension: locality in axiomatic QFT

⇒ Locality in the Haag-Kastler axioms means that algebras living inspacelike separated regions commute, as follows:

Axiom (Haag-Kastler Locality)

Given an algebra A(O) defined over a spacetime region O ⊂ M, withO′ ⊂ M denoting the set of spacetime points spacelike separatedfrom every point in O and A′ the set of operators that commute withevery operator in A (the ‘commutant’ of A), then

A(O′) ⊆ A(O)′.




Algebra of complement as subalgebra of commutant

)'O(A )'O(A

i´ M;h

'O 'OO

)O(A




Proof by construction?

⇒ Does ‘relativistic’ QFT give us a proof by construction thatquantum physics and SR are compatible?

Superficially, it certainly seems so:1 Klein-Gordon and Dirac equations are relativistically

invariant2 quantum states transform under Lorentz transformations as

they should3 conditions such as Haag-Kastler Locality demand that

superluminal action is ruled out




Tim Maudlin: claims like (3) are “empty” because...

Tim Maudlin. Quantum Non-Locality and Relativity. Blackwell Publishing (22002, 32011).

Spatial separation would be a per-fect insulator from causal influ-ences if no influence could go fasterthan light. Since Aspect’s experi-ments shows no such insulating ef-fects, some direct causal connec-tions must exist between space-likeseparated events. (64) ... [R]eliablereproduction of the quantum statis-tics demands superluminal causa-tion and superluminal informationtransformation. (194)




A sufficient condition for causation: counterfactual support

One immediately wants to ask: based on what analysis ofcausation does Maudlin make these pronouncements?

As he insists (126), a full analysis is unnecessary for presentpurposes; according to him, it suffices to have the followingsufficient condition for there to be causation:

ConditionGiven two local physical “events A and B, if B would not haveoccurred had A not occurred (or if B would have been different had Abeen different) then A and B are causally implicated with each other.”(128) (Although one can’t absolutely determine one to be the ‘cause’and the other the ‘effect’—they are spacelike separated after all.)

only ‘first approximation’: relational nomic connections excluded(to avoid counting ‘Cambridge change’ as causal implication)

questionable: really sufficient if no direction of causal influencecan be determined–even in principle?




Giving short shrift to Maudlin

In his detailed analysis of the possibly causal connection in Bellcorrelations in Chapter 5, Maudlin seems to make much weakerclaims than either before or after that chapter: the connection isonly one of “causal implication”, not causation.

(Causal implication may obtain between events neither of which is the cause ofthe other; cf. 129)

dilemma: either “causal implication” significantly weaker thancausation or not

If it is, then it arguably doesn’t violate SR, but it’s unclearhow this could satisfy our appetite for a causal explanationof the correlations; so it’s harmless, but impotent.If it is not, then that thirst is quenched, but on pains ofviolating SR.

⇒ either we disappoint our causal intuitions or we reject SR

So let’s see just how much we can get...




Analyzing factorizability

Jeremy Butterfield. Annals of the New York Academy of Sciences 755 (1995): 768-785.

Butterfield (1995): Landau, Summers and Werner haveestablished (between 1985 and 1988) that the Bell inequality isgenerically violated in AQFT.

In fact, there is a perfectly respectable realist approach to Bellcorrelations in non-relativistic QM, which gets encoded inaxiomatic QFT—albeit not in terms of causally efficacious localbeables.

To see this, consider the usual EPR-Bohm setup with a singletstate: |ψ(12)〉 ∝ | ↑(1)〉 ⊗ | ↓(2)〉 − | ↓(1)〉 ⊗ | ↑(2)〉

Main premise to derive Bell inequality:




Factorizability condition

p(12)ψ (xL, xR |L,R) = p(1)

ψ (xL|L) · p(2)ψ (xR |R),

where L and R are the measurement settings and xL and xR theoutcomes for the two wings, respectively.

Factorizability is equivalent to the conjunction of two conditions:

Outcome independence (OI)

p(1)ψ (xL|L,R, xR) = p(1)

ψ (xL|L,R)

p(2)ψ (xR |L,R, xL) = p(2)

ψ (xR |L,R)

Parameter independence (PI)

p(1)ψ (xL|L,R) = p(1)

ψ (xL|L)

p(2)ψ (xR |L,R) = p(2)

ψ (xR |R)




Axiomatic QFT: violating OI

Miklós Rédei and Stephen Summers, Foundations of Physics 32 (2002): 335-355.

As Butterfield (1995) shows, PI is encoded in Haag-KastlerLocality (or some similar axiom).

⇒ violation of OI, i.e. of ‘separability’

Important: OI and PI are statements of probabilisticindependence, their denial falls short of a commitment to acausal dependence.

It seems as if Bell’s theorem in itself does not entail that there beSR-violating causation.




The dynamical threat: the MP

Maudlin insists that “these features [Lorentz invariance,non-separability?] could secure the credentials only of a theorywhich has no wave collapse.” (194)

I take him to be saying that even if you accepted that thenon-locality is taken care of in QFT, dynamically the relativisticcompliance of QFT is not assured.

In other words, a relativistically accepted solution of the MP isrequired!

And the MP certainly rears its face in QFT...




Taking measureConsider a measurement setup in the rest frame of a lab:

post-measurement

pre-measurement

??

+

SR OR x

0t

1t

t




The MP in QFTJ Barrett. Wigner’s friend and Bell’s local field beables. Electronic J of Theoretical Physics. Forthcoming.

field F : both state of system S in spatial region RS and state ofobserver O in spatial region RO

|ψS0 (t)〉: state of F is zero in RS at t|ψS

+(t)〉: state of F is nonzero in RS at tmutatis mutandis for region RO (subscript r for ready state)

interaction Hamiltonian between local fields such that

|ψOr (t0)〉 ⊗ |ψS

0 (t0)〉 → |ψO0 (t1)〉 ⊗ |ψS

0 (t1)〉|ψO

r (t0)〉 ⊗ |ψS+(t0)〉 → |ψO

+(t1)〉 ⊗ |ψS+(t1)〉

Suppose field state in RS at t0 is |φS(t0)〉 = α|ψS0 (t0)〉+ β|ψS

+(t0)〉.unitary dynamics⇒ field state in RS and RO at t1 will be

α|ψO0 (t1)〉 ⊗ |ψS

0 (t1)〉+ β|ψO+(t1)〉 ⊗ |ψS

+(t1)〉




But in standard understanding, field state in RO must beseparable from state in RS after the interaction for there to be adeterminate record of the observer’s measurement, i.e. it musteither be |ψO

0 (t1)〉 (with probability |α|2) or |ψO+(t1)〉 (with

probability |β|2).

The moral is that insofar as there aredeterminate physical measurementrecords, if the quantum-mechanicalstate is taken to be complete, thenfield theory gets the dynamics of mea-surement wrong; and insofar as thereare determinate physical measure-ment records and the unitary dynam-ics is right, the quantum-mechanicalstate cannot be complete. (3)




A solution to the MP with relativistic credentials?

So, the MP must be addressed in QFT as well. And this is thedynamical threat to the compatibility between quantum physicsand relativity: to solve the MP without violating SR.

A solution to the MP must also offer an account of the Bellcorrelations, e.g.

Is OI or PI violated, and how?What is the causal goings-on in a EPR-Bohm situation?Are there causal arrows at all, and if so, are they directed?

This can only be denied on pain of rejecting realism.

MP in non-relativistic QM à la Maudlin (1995): either collapsetheory or hidden-variables theory or many-worlds theory

Tim Maudlin. Three measurement problems. Topoi, 14 (1995): 7-15.




MP in QFT/relativistic QM1 collapse:

only proposal is rGRWf (Tumulka 2006)is inchoate, so far no standard model—or even interactions (aproblem for AQFT too)

2 hidden variables:some field theoretic attempts (Bohm & Hiley; alt: Goldstein et al.)source of difficulty: Bohmian mechanics picks poison of causalmechanism over compatibility with SRBohmian mechanics violates PI and maintains OI, (axiomatic)QFT the other way around

3 many worlds:most promising: seems to only involve elements which can bemade Lorentz invariantproblem: not that it’s ontologically profligate, but to show howsemblance of collapse or indeterministic statistics arises (how dowe get probabilities, Born rule)

R Tumulka. A relativistic version of the Ghirardi-Rimini-Weber Model. Journal of Statistical Physics 125(2006): 821-840.


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Relativistic Quantum Physics