Transcript The Klein-Gordon Equation as a time-symmetric

The Klein-Gordon Equation Revisited
Ken Wharton
Associate Professor
Department of Physics
San José State University
San José, CA; USA
PIAF-1
February 1-3, 2008
Sydney, Australia
The PIAF Connection
• I will outline a foundational research program
that naturally links to very different work by at
least three PIAF participants:
– Robert Spekkens
– Huw Price
– Lucien Hardy
• If successful, this program should also interest:
– PI’s quantum gravity experts
– Australia’s retrocausal experts
– Bayesians (?)
The Big Picture
(for neutral, spinless fields)
General
Relativity
Special
Relativity
C
Klein-Gordon
Quantum Gravity
in curved space
Klein-Gordon
Equation
B
??
??
??
Quantum
Field
Theory
A
NonRelativistic
Limit
Schrödinger
Equation
Classical
Quantum
Mechanics
Quantum

Schrödinger’s starting point:
The Klein-Gordon Equation (KGE)
deBroglie Waves:  (x,t) ~ cos(k  x  t) ; E  , p  k
 2  2

2 2 2
2 4
 c   m c  (r,t)  0

2
 t

E  p c m c 0
2
2 2
2 4
 Particle
Relativistic
Klein-Gordon Equation (KGE)
Advantages: Time-Symmetric,
Relativistically Covariant

Problem: No consistent, spatially meaningful interpretation
General solutions to KGE:
(r,t)   a(k)ei(krt )  b(k)ei(krt )dk
(k)  k 2c 2  m2c 4 /
2
What happens in the non-relativistic limit?
One does NOT get the Schrödinger equation!
(1st and 2nd order differential equations
aren’t equivalent in ANY limit.)
(k)  k c  m c /
2 2
2 4

2
mc 2
k2

  0  1 (k)
2m
(r,t)   ei t a(k)ei(kr t )  ei tb(k)ei(kr t )dk
0
1
0
1

Schrödinger’s critical
assumption:

Then
i 0 t
(r,t)  e
(r,t)
where
b(k)  0
2

i

 2
t
2m
By dropping half of the allowed
 parameters, Schrödinger
reduced
the KGE to a 1st order differential equation (in t).


The critical assumption, in detail
 2  2

2 2 2
2 4
 c   m c  (r,t)  0

2
 t

The Klein-Gordon Equation
2
 

2

  (r,t)  0
i
 t 2m 
The Schrödinger Eqn. (V=0)

• Halves number of free parameters
in the solution.
(no longer need  and d/dt to solve; just )
• Introduces an explicit time-asymmetry.
• Arbitrary way to halve solutions (Asin+Bcos, A+iB, etc.)
• This particular halving fails in curved spacetime!
(Perhaps why QM has never been reconciled with GR)
It’s long past time to revisit the KGE!
The fathers of quantum mechanics never meant to devise a
relativistically-correct theory... and yet we’re still using their
basic formalism as a starting point 80 years later.
If relativistic thinking was irrelevant, then extrapolating to SR
or GR would be simple. The fact that it isn’t easy strongly
implies that SR+GR have foundational relevance to QM.
Ambitious Research Goal:
Use KGE to re-derive QM probabilities (associated with
preparation-measurement pairs) without dropping b(k).
(i.e. learn how to quantize a second-order differential equation)
The KGE’s “Extra” Free Parameters
The Klein-Gordon equation has a solution  with exactly twice
the free parameters of the Schrödinger Equation solution 
But 80 years of experiments say we can’t learn any more
information (at one time) than can be encoded by .
(Deeply connected with the Uncertainty Principle)
Therefore, if we start with the KGE as the master Equation,
one gets the axiomatic foundation of Spekkens’s toy model!
(We can only know half the total information in .)
However, this means one can never get enough information
to solve the KGE as an initial boundary condition problem.
Initial Boundary Conditions vs. CPT
Both quantum field theory and relativity are CPT symmetric;
should reduce to a time-symmetric non-relativistic picture.
But QM is explicitly time-asymmetric. (The T-asymmetry in
the Schrödinger Eqn is fixed “by hand”, the collapse is not.)
Connection with Huw Price’s work: Asymmetries appear
because boundary conditions are imposed asymmetrically.
To replace the T-asymmetric “collapse” with a
CPT-symmetric picture, maybe we shouldn’t be looking
for initial boundary conditions in the first place!
Boundary conditions are often
implemented time-asymmetrically
Atom A emits a photon, and it is later absorbed by atom B:
A
B
Very symmetric.
Using only initial boundary conditions leads to a strange picture:
B No time-symmetry
in this picture!
Upon reaching B, the rest
of the wave “collapses”?!
A symmetric picture
requires two-time
boundary conditions.
CPT and KGE: Natural Partners
Larry Schulman has attempted to impose two-time
boundary conditions on the Schrödinger equation.
“Time’s arrows and quantum measurement”,
L.S. Schulman, Cambridge Univ. Press (1997)
Leads to an overconstrained
equation; non-exact solutions.
But the Klein-Gordon equation requires a 2nd boundary
condition to determine the “extra” free parameters…
… it can’t go at the beginning, and physical timesymmetry implies it’s far more natural to put it at the end!
A Novel Proposal: Keep the full Klein-Gordon equation.
Impose half the boundaries at one time, and half at another time.
Mapping two-time boundaries to QM
Mathematical boundary conditions correspond to
external physical constraints (i.e. measurements).
Final measurement (procedure + results); allows retrodiction.
t=t0
(x,t)
Time
t=0
Initial measurement (preparation) can’t specify a unique wavefunction.
(Would need both (x,0) and

(x,0).)
t
If the boundary conditions correspond to measurements, the
“collapse” becomes the continuous effect of a future boundary.

Two-Boundary FAQs
Huw Price’s picture
of a photon passing
through 2 polarizers
45o “+” 0o
45o
? 0o
Doesn’t this violate our intuitive notion of causality?
Yes -- perhaps a benefit in disguise.
(Intuition is biased against time-symmetry)
Does this permit causal paradoxes?
It’s impossible to retrieve any future-information
without changing the boundary conditions.
Where does probability fit in?
Discrete Probability Weights
The 2-boundary problem is solvable, but cannot predict.
Furthermore, once you retrodict the solution , what
sense is there to extract an outcome probability from ?
Bayesian answer: “Probability is assigned to propositions,
not wavefunctions!”
Fact: Some pairs of boundaries are more likely to
occur together than other pairs of boundaries.
If relative weights for each pair are known, one can
generate probabilities for any time-biased proposition.
A Classical Example
Last semester, did a given student come to class for
two consecutive lectures?
yes
no
yes
no
5%
5%
0%
yes
no
no
yes
no
yes
no
20%
35%
35%
10%
yes
yes
no
no
Student 90%
“A”
yes
Student
“D”
time
time
Recovered probabilities: If A and D came to previous class,
A had a 94.7% attendance probability, while D had 36.4%.
Implementation Questions
This research program comes down to 2 main issues:
• What mathematical boundary condition corresponds to
a given physical measurement/constraint?
- Map to existing measurement theory?
- Construct GR-friendly measurement theory?
• What is the discrete probability weight that
corresponds to any complete solution?
- Demand exact correspondence to QM in NR-limit?
- Use known results as a guide, not a rule?
Recent Results (arXiv:0706.4075)
Standard theory: Boundary conditions are eigenfunctions
of an operator. (in position space, Xˆ  x, Pˆ  i  )
Problem #1: Pˆ  i  fails for the KGE!
 i(krt )
t )
(r,t)   a(k)ei(kr

b(k)e
dk



Propagates in k direction
Propagates in -k direction
Pˆ eigenvalues of both terms are k, which does not
correspond to physical momentum of the wave
2
Tentative solution: Use only time-even operators Q( Xˆ , Pˆ )

First Attempt: Two-time Boundary Conditions
 (r,t)  ?
Initial Boundary
Condition = F(r)
t=0
IBC:
FBC:
Final Boundary
Condition = G(r)
t = to
(r,t  0)  F(r)   F(k)eikrdk

(r,t  t0 )  G(r)   G(k)eikrdk
Fourier-expand
F(r) and G(r)
Plug into  (r,t) and solve for coefficients ak,bk.


F(k)e it  G(k)
a(k) 
e it  eit

0
0
0
F(k)eit 0  G(k)
b(k) 
eit0  e it0

a and b determine ; we know F(k) from initial boundary,
2 2
2 4
2


(k)

k
c

m
c
/
G(k)
from
final
boundary,
and
, but…

Next problem: infinite poles
a(k) 

it 0
F(k)e  G(k)
e it 0  eit 0
F(k)eit 0  G(k)
b(k) 
eit0  e it0
Problem #2;  is a function of k, so for any value of to,
there will always be values
of k where  t 0  n,

and the coefficient denominators go to zero!
Import the solution from quantum field theory:

give the mass a tiny imaginary component.
New KGE:
 2  2

2 2 2
2 4
 c   m c  i  0

2
 t

Then calculate probability and take limit as  0.

The “retrodicted” wavefunction
F (k)e(i  )t 0  G(k)
a(k)  ( i  )t 0
e
 e(i  )t 0


F(k)e(i  )t0  G(k)
b(k)  (i  )t0
e
 e (i  )t 0
(r,t)   a(k)ei(krt )et  b(k)ei(krt )etdk

• No Collapse ( automatically conforms to the final
boundary condition)
• Not pre-dictable: need measurement result G(r)
(Explains EPR/Bell w/o faster-than-light influences)
In other words, this is a “hidden variable” model that violates Bell’s
inequality, because the parameters a(k) and b(k) depend on future events.
Covariant Probability Weight
  * 
(r,t)  2 Im

mc
 t 
Charge density of KGE:
(not well-defined in curved space)
ct
FBC 
 
t=to
t=0
x
IBC
Here  is a unit four-vector, perpendicular to the boundary
condition’s 3D hypersurface (inward pointing).
Covariant generalization on
arbitrary closed boundary:
  * 
W
Im


m c BC   
Discrete Probability Postulate
Given by square of range of W:
(Wmax-Wmin)2 = P
W has a range because we don’t know the relative phase
between F and G, and we don’t know the exact value of to
Given: 1) Non-relativistic limit
2) Additional time-energy constraint
P  P0
(but not quite!)
Known non-relativistic limit:
P0 (F,G,t0 ) 
 F(k)G (k)e
*
it 0
3
d k
2
Four postulates: 3 good, 1 bad
• 1) Start with the Klein-Gordon Equation.
(Not the Schrödinger Equation!)
• 2) Constrain with a closed boundary condition in 4-D.
(Deal with infinities using m2 => m2-i)
2
* 


• 3) Weight the probability with P    Im

 
 BC
All of these postulates are easily extendible to a
general relativity framework (curved space), except…
 corresponds to the eigenstate
• 4) The boundary condition
from ordinary quantum measurement theory.
A spacetime view leads to a new
perspective of measurements
Standard View:
Spacetime View:
Measurement
time

Spatial boundary
conditions
time

hypersurface
boundary
condition
Preparation
space
The preparation and spatial
boundaries give , from
which one calculates the
measurement probabilities.
space
Partial information on a
hypersurface constrains the
solution . More solutions
lead to a larger weight P.
Physical interactions determine shape
and content of boundary conditions
Time
System
Lab+System
Space
Further insight can be found in recent papers:
R. Oeckl: “General Boundary Quantum Field Theory”: arXiv.org/hep-th/0509122
L. Hardy: “Non-Fixed Causal Structure”: arXiv.org/gr-qc/0608043
Clues to a GR-friendly measurement theory
• Momentum is not fundamental for fields in GR:
- The stress energy tensor, T, is fundamental.
• On a closed 3-surface (with dual ), one can extract:
- Energy density everywhere on surface: T0 
- Momentum density everywhere on surface: Ti 
 These appear to roughly map to the info in (x,t).
• On a space-like 3-surface, one can integrate the above
values to get total energy, angular momentum, etc...
The missing piece of the puzzle...
Quantization!
• Without eigenfunction rule, all possible boundary
conditions become reasonable.
(T00(x) need not be localized;  is a scalar field)
• Possible paths forward:
- Find probability weight that effectively selects for
eigenfunctions.
- New GR-friendly axiom: No paradoxes allowed.
A
B
(x,t)
A,B are space-like separated, but
can have a causal effect via 
Conclusions
• Relativity and CPT symmetry must inform quantum
foundations research, even in the non-relativistic limit.
• Both foundations and quantum gravity could benefit
from a new interpretation of the Klein-Gordon Equation
and a spacetime picture of measurement/boundaries.
This is a hugely ambitious research program...
...but PIAF is the group with the abilities and
research inclinations best suited to carrying it out.
Acknowledgements
Thank you to:
- Huw Price, Guido Bacciagaluppi, Centre for Time
- Jerry Finkelstein, Lawrence Berkeley Laboratory
- Eric Cavalcanti, Griffith University, Australia
- Philip Goyal, Perimeter Institute, Canada
More information can be found in these papers:
K.B. Wharton, “Time-symmetric quantum mechanics”, Foundations
of Physics, v.37 p.159 (2007)
K.B. Wharton, “A novel interpretation of the Klein-Gordon
Equation,” arXiv:0706.4075 [quant-ph]
Email: [email protected]