Ambiguity of Born’s rule in NLQM

Chapter IV: Different interpretations of quantum mechanics make different

4.3 Ambiguity of Born’s rule in NLQM

is clearly independent of Aa. We’ve shown that p(β|Aa,Bb) = p(β|Bb), so sQM doesn’t violate the no-signaling condition.

where, under some non-linear dynamics, |U_iii is the time-evolved |ii and U^†

f f E

is the backwards time-evolved |fi. The superscript NL explicitly indicates that NLQM is being used. Moreover, the proportionality sign in p_i←^{N L}_f follows from Í

f

D

i|U^†_f f E

2

being not, in general, normalized to unity. p_i→^{N L}_f and p_i←^{N L}_f are not necessarily equal, and so Born’s rule cannot be uniquely extended to NLQM.

4.3.2 Ambiguity in the boundary condition driving the non-linear time evo- lution

By extending Eq. (4.4) to NLQM, we can extend Born’s rule, given by Eq. (4.1), to NLQM. However, because NLQM is non-linear, a time-evolution operator doesn’t exist. Nonetheless, inspired by the state-dependent Heisenberg picture introduced in [6], we will show that we can define aboundary-dependent time-evolution operator, and that the choice of a boundary condition is the essential degree of freedom for extending Born’s rule to NLQM.

For some theories in NLQM1, running time-evolution requires solving the non-linear Schroedinger equation which contains a linear term, ˆHL, and a nonlinear term ˆVN L:

i~∂t|ψi=

HˆL+VˆN L(ψ(x,t))

|ψi. (4.14)

VˆN L(ψ(x,t))is a shorthand for a non-linear potential that depends on the wavefunc- tion |ψ(t)i expressed in some (possibly multi-dimensional) basis x. For instance, the Schroedinger-Newton equation for a single non-relativistic particle of mass m interacting with its own gravitational field is given by

i~∂t|ψi=

HˆL−m²G

∫

d³x|ψ(x,t)|²

|xˆ −x|

|ψi, (4.15)

whereψ(x,t)is the state |ψiexpressed in the position basis|xi.

We chose to write the nonlinear Schroedinger equation in the form of Eq. (4.14) to illustrate its similarity to the standard Schroedinger equation. Once we specify the boundary conditions, we can have Eq. (4.14) be formally equivalent to a linear Schroedinger equation. We will assume that a single Dirichlet boundary condition is sufficient to solve Eq. (4.14), and that its solution with the boundary condition

1As mentioned in the introduction, we will not investigate all possible non-linear theories. In particular we only consider theories with the form of Eq. (4.14), and whose solution can be uniquely specified with one Dirichlet boundary condition.

|ψ(T)i = |φiis|ϕ(t)i. Consequently, the linear Schroedinger equation i~∂t|ψi=

HˆL +VˆN L(ϕ(x,t))

|ψi (4.16)

is formally identical to Eq. (4.14) with the boundary-condition |ψ(T)i = |φi. Heuristically, in the context of Eq. (4.16), we can interpret ϕ(x,t) as a time- dependent classical drive. Eq. (4.16) has a time-evolution operator associated with it, which we denote by ˆUφ(T). The subscript is to emphasize that the time-evolution operator is associated with the boundary condition|ψ(T)i = |φi. We can now write the solution to Eq. (4.14) as

|ψ(t)i =Uˆφ(T)(t,T) |φi. (4.17)

We are ready to present the extension of Eqs. (4.1) and (4.4) to NLQM:

p^{N L}(α, β|Aa,Bb)= 1

N hα, β|α, βi, (4.18) where if Alice and Bob’s measurement events are spacelike separated

Ψ_c|α,β

= PˆβAˆ_φA

2(^T₂^A) (t₂,t₁)Bˆ_φB

2(^T₂^B) (t₂,t₁)PˆαUˆ_φ1(T₁)(t₁,t₀) |Ψ_inii, (4.19) andN = Í

α,βhΨ_c|α, β|Ψc|α, βi ensures that Í

α,βp^{N L}(α, β|Aa,Bb)is normal- ized to unity. Moreover, ˆUφ₁(T₁)(t1,t0)is the time-evolution operator fromt0 tillt1

and is associated with the boundary|ψ(T1)i = |φ1i. ˆA_φ^A

2(T₂)(t2,t1)( ˆB_φ^B

2(T₂)(t2,t1)) is the time-evolution operator associated with the boundary condition|ψ(T₂^A)i = |φ₂^Ai (|ψ(T₂^B)i = |φ^B₂i), and acts on Alice’s (Bob’s) particle from t1 till t2. The time- evolution of Alice and Bob’s joint system fromt1tillt2is separable because Alice’s particle’s future light-cone att1does not overlap with Bob’s particle’s past light-cone att2. As a result, their total interaction Hamiltonian, which includes contributions from the linear Hamiltonian ˆHLand from the non-linear potential ˆVN L, must be zero.

Note that, by construction, Eq. (4.19) recovers the predictions of sQM when the non-linearity ˆVN L vanishes.

If the events A and B were time-like separated, then there are numerous schemes for extending the time evolution of Alice and Bob’s particles to NLQM. Call the solutions to Eq. (4.14) with the boundary conditions |ψ(T₂^A)i = |φ₂^Ai, |ψ(T₂^B)i =

|φ^B₂i and |ψ(T₂^AB)i = |φ₂^ABi as |ϕ^A(t)i, |ϕ^B(t)i and |ϕ^AB(t)i respectively. We can

then write the non-linear potential ˆVN L in Eq. (4.16) in the following general way:

VˆN L =VˆA

ϕ^A(x,t)

⊗ IˆB+IˆA⊗VˆB

ϕ^B(x,t) +Vˆint

ϕ^AB(x,t)

, (4.20) where ˆVA( ˆVB) is the free non-linear Hamiltonian acting on Alice’s (Bob’s) particle, and ˆVint is the non-linear interaction potential. However, we find it difficult to justify why each term in ˆVN L would be generated by a different boundary condition when Alice and Bob’s particles are allowed to directly communicate and interact. We will imposeφ₂^A = φ₂^B = φ₂^AB andT₂^A =T₂^B = T₂^AB when the measurement events A and B are timelike separated. We summarize our chosen form of

Ψ_c|α,β by

Ψ_c|α,β

=











PˆβUˆ_φ2(T₂)(t₂,t₁)PˆαUˆ_φ1(T₁)(t₁,t₀) |Ψ_inii A & B timelike, PˆβAˆ_φA

2(T₂^A) (t₂,t₁)Bˆ_φB

2(T₂^B) (t₂,t₁)PˆαUˆ_φ1(T₁)(t₁,t₀) |Ψ_inii A & B spacelike.

The introduction of arbitrary boundary conditions|φ₁iatT₁and|φ₂iatT₂might ap- pear artificial, but isn’t. Each formulation of quantum mechanics predicts different boundary conditions after a measurement. For instance, in Eq. (4.14), an inter- pretation of quantum mechanics with wavefunction collapse states that |φ₁i = |αi andT₁= t₁, while the Everett interpretation states that|φ₁iis the initial state of the universe andT₁ is when the universe began. Refer to section 4.1 for more details.

In sQM, we do not have to worry if and how the wavefunction collapses because the time-evolution operator is well-defined independently of the wavefunction it acts on.

However, in NLQM, each boundary condition generates a different time-evolution operator, and so how we formulate quantum mechanics matters in NLQM.

4.3.3 Extending the formalism to relativistic quantum mechanics

We can rigorously study superluminal communication only in quantum field theory, where the total Hamiltonian consists of free and interaction (between different fields) energy densities

Hˆ =

∫

d³xHˆ₀(x)+

∫

d³xHˆint(x). (4.21) Assigning spatial locations for quantum degrees of freedom is crucial for placing constraints on ˆHto ensure that it is causal. Let ˆH_intbe the interaction energy density in an interaction picture with respect to∫

d³xHˆ₀(x), and then ˆH_int commutes over spacelike distances [20]

hHˆ_int(tx,x), Hˆ_int t_y,yi

= 0, c tx−ty2

− |x−y|² < 0. (4.22)

We generalize ˆH to include a dependence on a wavefunction:

Hˆ^{N L}(t)=∫

d³xHˆ₀ x,

Ψ_Φ(T)(t) +∫

d³xHˆint x,

Ψ_Φ(T)(t) , (4.23) where

Ψ_Φ(T)(t)

is the solution to the non-linear Schroedinger equation i~∂t|Ψ(t)i = ∫

d³xHˆ₀(x,|Ψ(t)i)+∫

d³xHˆint(x,|Ψ(t)i)

|Ψ(t)i (4.24) with the boundary condition |Ψ(t =T)i = |Φi. We further generalize ˆH^{N L} by allowing for different boundary conditions at each location

Hˆ^{N L}(t)=

∫

d³xHˆ0 x,

Ψ_ΦT(x)(t) +

∫

d³xHˆint x,

Ψ_ΦT(x)(t) , (4.25) where

Ψ_ΦT(x)(t)

is the solution to Eq. (4.24) with boundary condition|Ψ(t =T(x))i =

|Φ(x)i.

The relativistic non-linear generalization of Ψ_c|α,β

in Eq. (4.2) is Ψ_c|α,β

= PˆβUˆ_Φ⁽1)

T (x)(t2,t1)PˆαUˆ_Φ⁽0)

T (x)(t1,t0) |Ψ_inii, (4.26) where x ∈ R³, and ˆU_Φ⁽⁰⁾

T(t,x) ( ˆU_Φ⁽¹⁾

T(t,x)) is the time-evolution operator associated with the boundary condition

Ψ

t =T⁽⁰⁾(x) E

=

Φ⁽⁰⁾(x) (

Ψ

t =T⁽¹⁾(x) E

= Φ⁽¹⁾(x)

).