QM's Classical Inheritance

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Foundations of Quantum Mechanics

QM’s Classical Inheritance Series Outline

1. How to be a Quantum Mechanic

2. Entanglement, measurement and decoherence

3. A Gordian knot and Heisenberg’s cut

4. Local hidden realism: Einstein’s “reasonable” solution

5. QM’s classical inheritance

6. Bohmian realism: non-local hidden variables and holism

7. How many cats does it take to solve a paradox? Classical Mechanics

There are three main formulations of classical mechanics:

• Hamiltonian mechanics

• Lagrangian mechanics

• Hamilton-Jacobi mechanics

Each formulation is equivalent and all have pros and cons when describing a physical system

QM is implicitly constructed by “quantizing” these formulations

can we make connections between CM and QM in a formal sense? CM to QM

Each formulation of classical mechanics corresponds to a formulation of QM:

Hamiltonian Weyl-Wigner QM

Lagrangian path integral QM

Hamilton-Jacobi pilot wave QM Limits

If a theory is a generalization of another there should exist a well deﬁned limit in which it reproduces the results of the old theory

! old theory

! new theory

! e.g. special relativity v we have the dimensionless deformation parameter = c non-relativistic limit given by 0 ! 1 2 ! = 1+ + (4) 1 2 ! 2 O ! p = mv mv + (2) p ! O 1 T =(1 )mc2 mv2 + (4) ! ! 2 O

Special relativity is fundamentally different to classical mechanics:

• Minkowski space-time vs Cartesian product of space and time

But framework (objectively real unique objects, determinism, etc) the same

• we recover the behaviour of classical objects in the non-relativistic limit Hamiltonian Mechanics

Deﬁne a “phase space”, T*(C), with position and momentum coordinates

! (q, p)

The phase space has a “symplectic” structure which deﬁnes a “Poisson bracket”

! @f @g @g @f f,g = { } @q @p @q @p !

• a classical particle is characterized by a point in phase space

• time evolution of observables driven by Hamilton’s equations

! @H @H q˙ = , p˙ = @p @q The Hamiltonian induces a vector field on the phase space @H @ @H @ ! H, = { } @q @p @p @q Integral curves of this field define a set of trajectories parameterized by “time”

! (t)=(q(t),p(t))

Observables vary along these trajectories (in “time”) according to dA @A ! = A, H + T*(C) dt { } @t in particular @H @H q˙ = , p˙ = ! @p @q

(mathematically this is the Lie derivative with respect to the Hamiltonian vector ﬁeld)

• quantities with zero Poisson bracket are conserved along trajectories

• classical particles with definite properties follow trajectories If we have an ensemble of particles we can define a probability density in phase space ⇢(q, p) ! with associated probability of finding a particle in dV

! P = ⇢(q, p) dnqdnp

This density satisﬁes a continuity equation and so evolves in time according to the Liouville equation @⇢ + ⇢,H =0 ! @t { }

• describes an incompressible ﬂuid of non-interacting particles

• expectation values obtained by averaging quantities, weighted by the probability distribution

A = ⇢(p, q)A(p, q) dnqdnp h i Z QM from deformed CM

Obvious differences between QM and CM:

• QM observables are operators with non-trivial commutation relations

• quantum states, and observables, live in a Hilbert space, not a phase space

• we can’t assign deﬁnite properties to any single system, only ensemble averages

Canonical (Weyl) quantization:

• promotes observables to operators, poisson brackets to commutators

i⇠(Qˆ q)+i⌘(Pˆ p) n n n n ! ˆ(f)= f(q, p) e d qd pd ⇠ d ⌘ O Z

classical observable group element in the Heisenberg Lie group How can we deﬁne the classical limit and recover CM?

• naively just take ~ 0, [Q,ˆ Pˆ]=i~ 0 ! ! • all operators can be replaced by numbers, but states still live in Hilbert space and they’re not necessarily localized!

• how do we connect this to classical mechanics?

Instead consider Wigner’s phase space representation of QM:

• we can map operators onto the phase space via the Wigner transform ⇠ ⇠ ! f(q, p)= q Oˆ(f) q + eip⇠/~ d⇠ 2 2 Z ⌧ • the Wigner function W ( q, p ) is the Wigner transform of the density operator • QM expectation values calculated from phase space integrals

Oˆ = W (q, p) O(q, p) dnqdnp h i Z Non-commuting operators can be accommodated in this framework by deforming the Poisson structure to a Moyal structure:

• deﬁne the Moyal product of two functions on the phase space

i ~ @ @! @ @! ! f ? g = fe2 x p p x g

• a non-commutative product of phase space functions

We can use this product to deﬁne a Moyal bracket for observables: 1 ! f,g = f ? g g ? f {{ }} i~ The Moyal bracket describes the time evolution of an observable df @f ! = f,H + dt {{ }} @t and generates a quantum version of the Liouville equation @W + W, H =0 @t {{ }} Now we can take the classical limit in phase space:

• expand the Moyal bracket in the deformation parameter

1 i~ 3 2 ! f,g = ( f,g g, f )+ (~ ) = f,g + (~ ) {{ }} i 2 { } { } O { } O ~ • we recover the Poisson bracket of classical mechanics!

Can we interpret the Wigner distribution as a classical Liouville density?

• in general W ( q, p ) can be negative (no probability interpretation)

• can display non-classical entanglement W (q ,p ,q ,p ) = W (q ,p )W (q ,p ) 1 1 2 2 6 1 1 2 2 We could coarse grain the phase space into cells or “blobs” of volume ~

• interpret W ( q, p ) as a classical ensemble of cells

• usually tends to an ignorance interpretable Liouville density as ~ 0 ! • some states still display entanglement in classical limit (c.f. quantum chaos) Lagrangian Mechanics

Deﬁne a tangent conﬁguration space T(C), with position and velocity coordinates (q, q˙)

The “action” is the time integral of the “Lagrangian”, (basically L=T-V) t1 ! S = dtL(q, q˙) Zt0 “Hamilton’s principle” says that the conﬁguration chosen by nature is the one that minimizes the action S =0 (qf , q˙f ) This leads to the Euler-Lagrange (E-L) equations of motion, d @L @L (t)=(q(t), q˙(t)) ! =0 dt @q˙ @q 1 e.g. if L ( q, q ˙ )= m q˙ 2 mgq then the E-L equations yield 2 T(C) 1 2 q˙(t)=q ˙0 gt q(t)=q0 +˙q0t gt 2 (q0, q˙0) A distribution of classical particles will evolve along the trajectories which minimize their action to produce a new distribution

! ⇢(t1)

! ⇢(t0)

• the Euler-Lagrange equations are satisﬁed along these trajectories Path Integral QM

The superposition principle means that we can relate wavefunctions at different points in space and time

! (x,t)= K(x,t; r,t0) (r,t0) dr Z The propagator, K, can be written as a weighted sum over all possible paths

! iS/~ t K(x,t; r,t0)= Dq(t) e S = dt0 L(q, q˙ ,t) Z t0 ! (x,t) Z

(r,t0) In the classical limit as,

! ~/S 0 ! many of the weights of the paths cancel in the path integral

! C eiS/~ ! S !

! q(t) only paths around the minimum of the action (classical path) positively interfere

! q(t)classical Does this provide a good classical limit?

• dynamics become dominated by classical action

• states are still wavefunctions, potentially non-classical states

! (x,t)= K(x,t; r,t0) (r,t0) dr Z • can prepare a semi-classical initial-state, but even these diffuse

What path is chosen if there are multiple stationary points?

• in CM a unique path is determined by initial-conditions ( q0, q˙0)

• in QM have to sum over all classical paths… even if ~ /S 0 e.g. the double slit experiment ! Hamilton-Jacobi Mechanics

Based on the Hamilton-Jacobi (H-J) equation, @S ! + H(q, p)=0 @t Where S is Hamilton’s Principle Function, closely related to the action @S 1 Conjugate momentum defined by p = , defines a velocity field v = S @q mr We have the H-J equation, @S 1 = ( S)2 + V ! @t 2m r

• the details of the potential deﬁne the solution S and the possible trajectories

• individual particles move along trajectories in conﬁguration space

An ensemble of particles has a distribution which satisﬁes a continuity equation @⇢ + (⇢v)=0 @t r · Pilot wave

Consider the polar form of the wavefunction iS(x,t)/ ! (x,t)=R(x,t)e ~ plugging this into the Schrödinger equation we get two coupled equations

! 2 2 2 2 @R 2 S @S ( S) ~ R + R r =0 = + V @t r · m r r ! ✓ ◆ @t 2m 2m R We recover a continuity equation and Hamilton-Jacobi equation if:

2 • probability of finding a particle at (x,t) ⇢(x,t)=R(x,t) 1 • velocity field defined as v(x,t)= S(x,t) mr • H-J equation has a potential modified by a “quantum potential” ~2 2R Q = r 2m R Classical H-J theory deals with an ensemble particles moving in a potential

• this allows a statistical description in terms of a wave on conﬁguration space, particle dynamics generates S… ensemble represented by a wave

Pilot wave ﬁelds derived from wavefunction on conﬁguration space

! R(x,t)= (x,t) ,S(x,t)=~Im ln( (x,t)) | | • wavefunction deﬁnes S which in turn inﬂuences dynamics⇥ , not passive⇤

• classical trajectories can cross in conﬁguration space, cannot in pilot wave quantization

• in the limit Q 0 , Q 0 classical H-J dynamics recovered ! r ! • ~ 0 not sufﬁcient as Q is state (and potentially ~ ) dependent ! • many non-classical states have Q=0 , e.g. free particle wavefunction

• classical limit requires ~ 0 and an appropriate selection of quantum states ! Decoherence

In all formulations, decoherence is necessary to derive a classical description

Coupling a system to a quantum environment:

• ensures the Wigner function is positive (kills interference)

• localizes physical states through scattering

• selects conﬁguration space as a preferred basis

• restricts allowed quantum states to set of classical-like states

But… only applies to open systems:

• implies that there is no well deﬁned classical “universe” as a limit of QM Summary

• There are three main formulations of classical mechanics

• our formulations of QM inherit many features from these constructions, even if they differ in detail and interpretation

• we can usually recover classical dynamics in the limit ~ 0 ! • the full classical limit also requires us to restrict the Hilbert space to a small subset of allowed states

• classical-like states arise only effectively from decoherence in open systems

We formulate QM employing many conceptual tools of CM (state, dynamics, etc), but there is no formal correspondence between the two

Is it possible to formulate QM starting from a completely non-classical perspective? Series Outline