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Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!


  

F  d  E
 
 V 
d
c
d
V
d
V
Paradox V
W. Shockley and R.P. James, PRL 171, 1370 (1967)
A cannon with no recoil
Paradox V
W. Shockley and R.P. James, PRL 171, 1370 (1967)
A cannon with no recoil
An isolated system consists of a current loop (two oppositely rotating and
oppositely charged disks) and a charge which are originally at rest.
When the current dies out, the charge starts moving, while the disks apparently
stay in place.
Resolution of Paradoxes V, IV,III
(Re)discovery of “hidden momentum”
Y. Aharonov, P. Pearle, and L. Vaidman, PRA 38, 1863 (1988)
A current loop in a static electric field has a nonzero linear momentum
Resolution of Paradox V (recoil-free cannon)
When the current stops, the hidden
momentum of the current loop is transferred
to the mechanical momentum of the tube.
The loop recoils to the left.
Resolution of Paradoxes V, IV,III
(Re)discovery of “hidden momentum”
Y. Aharonov, P. Pearle, and L. Vaidman, PRA 38, 1863 (1988)
A current loop in a static electric field has a nonzero linear momentum
Resolution of Paradox V (recoil-free cannon)
When the current stops, the hidden
momentum of the current loop is transferred
to the mechanical momentum of the tube.
The loop recoils to the left.
There is no recoil-free cannon.
Resolution of Paradox IV
 
F 0


dp
F 
dt
but
 
a 0
 

p  mV  phidden


dphidden exactly
F 
dt

 
a 0
Resolution of Paradox III
The motion of the electron inside the interferometer is the same with or without
the solenoid

ELECTRON
ELECTRON
AC dual to AB
F 0
 
a 0

NEUTRON
F 0
 
a 0
NEUTRON
LINE OF CHARGE
The motion of the electron is identical
to the motion of the neutron
The motion of the neutron inside the
interferometer is the same with or
without the line of charge.
Paradox I
At every place on the paths of the wave packets of the
particle there is no observable action, but nevertheless,
the relative phase is obtained.
Aharonov-Bohm Effect
ELECTRON
Aharonov-Casher Effect
NEUTRON
SOLENOID
LINE OF CHARGE
Conclusion
Paradox I is an unavoidable property of both Aharonov
Bohm and Aharonov Casher effects which makes them
nonlocal topological effects
How comes hidden momentum?
A current loop in a static electric field has a nonzero
linear momentum
Hint: paradox VI
Paradox of Two Lorentz Observers
Paradox VI
Charged particle, charged plate, and two Lorentz Observers
Alice’s view
Paradox VI
Charged particle, charged plate, and two Lorentz Observers
Bob’s view
Hidden momentum
The current loop model: free charges
moving inside a frictionless tube
Vi
(This and other models: L. Vaidman, AJP 58, 978 (1990))
q


 qiVi  0
p hid
i
i
 
 mi0ViVi i 00
i
ptot 
E
 miVi 
i

i


ptot  phidden
m0iVi
1
2
0
Vi
c2

E

c

Bohm versus Everett
30.08.2010
21st-century directions in de Broglie-Bohm theory and beyond
THE TOWLER INSTITUTE The Apuan Alps Centre for Physics
Vallico Sotto, Tuscany, Italy
Hope:
Big hope:
Today’s physics explains all what we see.
Today’s physics explains All.
Bohr (SEP): The quantum mechanical formalism does not provide
physicists with a ‘pictorial’ representation: the ψ-function does not,
as Schrödinger had hoped, represent a new kind of reality.
Instead, as Born suggested, the square of the absolute value of the
ψ-function expresses a probability amplitude for the outcome of a
measurement.
Bohr and today’s majority of physicists gave up the hope
I think, we should not.
Bohm and Everett are candidates for a final theory.
Bohm:
All is Ri
and 
Everett:
All is 
Everett:
All is
Many-Worlds
http://qol.tau.ac.il/TWS.html
The Quantum World Splitter
Choose how many worlds
you want to split by pressing
one of the red dice faces.
http://qol.tau.ac.il/TWS.html
left
right
http://qol.tau.ac.il/TWS.html
right
World-splitter of Tel Aviv University
A
B
World-splitter of Tel Aviv University
A
B
World-splitter of Tel Aviv University
A
B
All
All is a closed system which can be observed
All
All is a closed system which might include an observer
which can be observed
What is ψ ?
There is no sharp answer. Theoretical physicists are very flexible in
adapting their tools, and no axiomization can keep up with them.
But it is fair to say that there are two core ideas of quantum field
theory.
First: The basic dynamical degrees of freedom are operator
functions of space and time- quantum fields.
Second: The interaction of these fields are local in space and time.
F. Wilczek (in Compendium of Quantum Physics, 2009)
Bohm: At the end of the day, the only variables we observe are positions.
( A (r ),  (r ))
a
a
(r )
Space is taken for granted
Everett:
(r )
Bohm:
(r )


Bohm:
r
(
t
),
r
(
t
),....,
r
(
t
)


1
2
N
All is
and ( r1 , r2 ,...., rN , t )
evolving according to deterministic equations
Everett:
All is ( r1, r2 ,...., rN , t )
evolving according to deterministic equation
A CENTURY AGO:
All is particles
evolving according to Newton’s equations
 r1(t ), r2 (t ),...., rN (t )
Laplacian determinism
Laplacian determinism

TRIVIAL
Observation
 r1(t ), r2 (t ),...., rN (t )
Bohmian mechanics

TRIVIAL
Observation
 r1(t ), r2 (t ),...., rN (t )
Everett Interpretation

HARD
Observation
( r1 , r2 ,...., rN , t )
Laplacian determinism

TRIVIAL
Observation
 r1(t ), r2 (t ),...., rN (t )
Bohmian mechanics

TRIVIAL
Observation
 r1(t ), r2 (t ),...., rN (t )
Everett Interpretation

HARD
Observation
( r1 , r2 ,...., rN , t )
Laplacian determinism

TRIVIAL
Observation
 r1(t ), r2 (t ),...., rN (t )
Bohmian mechanics

TRIVIAL
Observation
 r1(t ), r2 (t ),...., rN (t )
Everett Interpretation
Many parallel
Observations

HARD
( r1 , r2 ,...., rN , t )
What is “a world” in the Everett Interpretation ?
Many parallel
Observations

many
worlds

world i

( r1 , r2 ,...., rN , t )
(r1, r2 ,...., rN , t )  i i ( r1 , r2 ,...., rN , t )
Observation i

An observer has definite experience.
 i   iOBSERVER iREST
Everett’s Relative State World
A world is the totality of (macroscopic)
objects: stars, cities, people, grains of sand,
etc. in a definite classically described state.
The MWI in SEP
i ( r1 , r2 ,...., rN , t )
i   iOBJECT1  iOBJECT2 ... iOBJECTK iREST
i
OBJECT
is a Localized Wave Packet
for a period of time
What is our world in the Bohmian Interpretation ?
Observation

 r1(t ), r2 (t ),...., rN (t )
We do not observe (experience) ( r1 , r2 ,...., rN , t )
Bohmian trajectories
CONTEXTUALITY
EPR
V
V
V V
V V
V
V
V
V
V
V
V
V
V
V
V
V
V
V V
V /3
V  1   V   0.5 V

1  0.5
3
V
V
V /3
V
V
V
V
V
V
MZI
IFM
Counterfactual Computation
0
Surrealistic trajectories
A tale of a single world universe
The king forbade spinning on distaff or spindle,
or the possession of one, upon pain of death,
throughout the kingdom
A tale of a single world universe
The king forbade performing quantum measurements, or
the possession of quantum devices, upon pain of death,
throughout the kingdom
The Quantum World Splitter
Photomultipliers
Geiger counters
Stern Gerlach devices
Beam splitters
Down conversion crystals
Quantum dots
Quantum tunneling
Photodiods
……
A tale of a single world universe
UNIVERSE  WORLD   OBJECT1  OBJECT2 ... OBJECTK  REST
Quantum states of all macroscopic objects are
Localized Wave Packets all the time
Zero approximation: all particles remain in product LWP states
WORLD ( r1 , r2 ,...., rN , t )   1 ( r1 )  2 ( r2 )... N ( rN )
n
( rn )
Particles which do not interact strongly with “macroscopic objects”
need not be in LWP states.
WORLD   1 ( r1 )  2 ( r2 )... K ( rK )  REST
Particles which make atoms, molecules, etc. can (and should
be) entangled among themselves. Only states of the center of
mass of molecules, cat’s nails etc. have to be in LWP states.
2
M
1
1
1
WORLD   CM
(r1CM ) rel
(r1i  r1 j )  CM
(r2CM ) rel2 (r2i  r2 j )... CM
(rMCM ) rel
(rMi  rMj )  REST
A tale of a single world universe
Quantum states of all macroscopic objects are
Localized Wave Packets all the time
UNIVERSE ( r1 , r2 ,...., rN , t )   1 ( r1 )  2 ( r2 )... N ( rN )
2
M
1
1
1
WORLD   CM
(r1CM ) rel
(r1i  r1 j )  CM
(r2CM ) rel2 (r2i  r2 j )... CM
(rMCM ) rel
(rMi  rMj )  REST
  (r )
 ( r ) of a cat!

TRIVIAL
Observation
 ( r1 )  ( r2 )... ( rN )
1
N
2
Almost the same as in




Bohmian trajectories













Two worlds universe
This is a multiple worlds universe
Two worlds universe
A
B
Two worlds universe
One world does not disturb the other
A
B
Two worlds universe
One world does not disturb the other
A
B
Two worlds universe
Preferred basis:
| A,| B or
 | A | B | A | B 
,

  | ,| 
2
2 

| A | R MD A | R MD B | R ENV
STABILITY
A
B
| A | V  MD A | R MD B | R ENV
| A | V  MD A | R MD B | A ENV
Two worlds universe
Preferred basis:
| A,| B or
 | A | B | A | B 
,

  | ,| 
2
2 

| A | R MD A | R MD B | R ENV
STABILITY
A
| A | V  MD A | R MD B | R ENV
| A | V  MD A | R MD B | A ENV
| B | R MD A | R MD B | R ENV
| B | R MD A | V  MD B | R ENV
| B | R MD A | V  MD B | B ENV
B
Two worlds universe
Preferred basis:
| A,| B or
 | A | B | A | B 
,

  | ,| 
2
2 

| A | R MD A | R MD B | R ENV
STABILITY
A
| A | V  MD A | R MD B | R ENV
| A | V  MD A | R MD B | A ENV
| B | R MD A | R MD B | R ENV
| B | R MD A | V  MD B | R ENV
| B | R MD A | V  MD B | B ENV
|  | R MD A | R MD B | R ENV
B

| 
| A | V  MD A | R MD B  | B | R MD A | V  MD B
2
| R ENV
| A | V  MD A | R MD B | A ENV  | B | R MD A | V  MD B | B ENV
| V  MD A | R MD B | A ENV  | R MD A | V  MD B | B ENV
2

2
 | 
| V  MD A | R MD B | A ENV  | R MD A | V  MD B | B ENV
2