Transcript Document

SEVEN DEADLY QUANTUM SINS
C. S. Unnikrishnan
Fundamental Interactions Laboratory
Tata Institute of Fundamental Research,
Mumbai 400005
tifr.res.in/~filab
The truncated list:
1.
Quantum Muddle  Identifying the physical system with the state/w-function
1b) What exactly is a Schrodinger Cat?
2.
Uncertainty principle, disturbances and measurements on single systems
3.
What exactly is the EPR argument as meant by Einstein?
4.
Bell’s inequalities and theories of correlations
5.
Do experimental results indicate (let alone ‘prove’) nonlocality?
Entanglement=Nonlocality?!
6.
The belief of the reality of zero-point modes in vacuum
7.
Black holes and information loss – serious misunderstanding of black hole
gravity
There are other ones…
1) (Fact )Present quantum theory requires the assumption that QM is
NOT applicable to all physical systems
2) (Belief) Decoherence solves the problem of ‘quantum to classical
transition’
3) …
1. THE QUANTUM MUDDLE
1. The Quantum Muddle
Quantum system = Quantum state, physically.
(Wavefunction = Particle and wavefunction has a space-time existence)
The notion arose from peculiarities of single particle quantum mechanics and
wave-particle duality.
 ( z )   1( z )  2( z )
‘Photon’ interferes with itself…
Particle is in both paths….

1
z  z
2

Of course, it will be a severe mistake if one interprets this state as the particle
being both up and down at the same time, because the state is

1
z  z
2
 x
An individual physical system in classical physics might be an element of an
ensemble with a probability distribution for its observables – however it will be
silly to identify the individual with the distribution.
Local detection, ‘Collapse of the wavefunction’ etc.
What is the ontological description one has in mind?
This of course leads to severe confusion of nonlocality. However the serious
problem has been in relation to Schrodinger cat and entanglement.
Quantum muddle and the Schrodinger cat:
1
 
Atom, Cat   Atom, Cat 

2
 cat
1

Cat   Cat 

2


Imagine describing an electron in a superposition of up and down spin
projection states and an electron spinning both up and down! Quantum
mechanically it has a spin projection orthogonal (geometric) to being up or
down.

1
 

2
1
 
2
1

2

Schrodinger would have been appalled to
know that this state of elementary particle
is also called by many as a S-cat!
2.
Uncertainty principle, disturbances and measurements on single systems
p  h /   h /  x    2
Suppose there was only ONE which-way sensor:
Interference is still washed out 100%! So, vanishing of interference and
coherence has nothing to do with disturbance of momentum transfer.
Can one measure the position and momentum of a particle more accurate than
allowed by the uncertainty relation ?
Source slit and detector resoltuion   x
What is the resolution of momentum determination?
After a large number of measurements, what is the position uncertainty?
3.
What exactly is the EPR argument as meant by Einstein?
Excerpts from Einstein’s letter to Popper (reproduced in Logic of Scientific
Discovery) explaining his view that the wave-function description is
incomplete:
“Should we regard the wave-function whose time dependent changes are, according to
Schrödinger equation, deterministic, as a complete description of physical reality, and
should we therefore regard the (insufficiently known) interference with the system from
without as alone responsible for the fact that our predictions have a merely statistical
character?
The answer at which we arrive is the wave-function should not be regarded as a complete
description of the physical state of the system.
We consider a composite system, consisting of the partial systems A and B which interact for
a short time only.
We assume that we know the wave-function of the composite system before the interaction
– a collision of two free particles, for example – has taken place. Then Schrodinger
equation will give us the wave-function for the composite system after the interaction.
Assume that now (after the interaction) an optimal measurement is carried out upon the partial
system A, which may be done in various ways, however depending on the variables which one
wants to measure precisely – for example, the momentum or the position co-ordinate. Quantum
mechanics will then give us the wave-function for the partial system B, and it will give us
various wave-functions that differ, according to the kind of measurement which we have chosen
to carry out upon A.
Now it is unreasonable to assume that the physical state of B may depend upon
some measurement carried out upon a system A which by now is separated from
B (so that it no longer interacts with B); and this means that the two different
wave-functions belong to one and the same physical state of B. Since a complete
description of a physical state must necessarily be an unambiguous description
(apart from superficialities such as units, choice of the co-ordinates etc.) it is
therefore not possible to regard the wave-function as the complete description of
the state of the system.”
Anything beyond this in the EPR Phys. Rev. paper is superfluous and irrelevant.
In particular there is no reference or wish regarding a possible completion of QM
using some classical statistical hidden variables.
4. Bell’s inequalities and theories of correlations
5. Do experimental results indicate (let alone ‘prove’) nonlocality?
Entanglement=Nonlocality?!
Beliefs:
1) Experimental results prove that there is nonlocality (violation of Einstein locality)
2) Local Hidden Variable Theories are theoretically valid (no inconsistencies)
From Wikipedia…as a frequent example
Experimentalists such as Aspect used this inequality[1], as well as other
formulations of Bell's inequality, to invalidate the hidden variables hypothesis
and confirm the existence of nonlocality in quantum mechanics, implying that
the lack of determinism in the Copenhagen interpretation was justified.
The case of two ‘spin-half’ particles:
1
S 
1 1 1 2  1 1 1 2 

2
a
b
S=0
A
1
P( a, b ) 
N
B
 i Ai Bi : Ai , Bi  1
Important input
Quantum Mechanics: P( a, b )   a  b   cos
P(a, b )QM   S  1  a   2  b  s  a  b
P(a, b )Bell   A(a, h)B(b , h)  ( h)dh
The essence of Bell’s theorem is that these two correlation functions have
distinctly different dependences on the angle between the settings of the
apparatus (difference of about 30% at specific angles).
correlation
angle
0
-1
A
B
A’
B’
A, B, A ', B '  1
Consider the quantity AB  A ' B  A ' B ' AB '
S (1, 2)  AB  A ' B  A ' B ' AB '  A( B  B ')  A '( B  B ')
Now, the assumption ‘reality’ is made
Either B  B '  S (1, 2)  2 A '
Or B  B '  S (1.2)  2 A
P(1, 2)  S (1, 2) ,  2  P(1, 2)  2  P  2
However S (1, 2)QM  2 2
So, what does the experimental confirmation of the violation of Bell’s inequality
imply as valid theoretical statements that are logically rigorous?
1) Quantum mechanics is validated as a good theory of correlations…
2) A classical hidden variable theory in which statistically distributed valued of
the HV determine measurement outcomes is validated as a possible good
theory of correlations provided there is violation of Einstein locality.
The common sin is to mix the two and claim that experiments prove nonlocality!
A reanalysis of what Bell did to get the inequalities:
PB ( a , b )    ( h )dh A( a , h ) B (b , h ),
  (h )dh  1
Since A( a )   B ( a ) and PB ( a , a )  1, Bell wrote
PB ( a , b )     ( h ) dh A( a , h ) A( b , h )

    ( h ) dh [ A( a , h ) A( b , h )  A( a , h ) A(b , h ) A(b , h ) A( c , h )]
   ( h ) dh A( a , h ) A(b , h )[ A(b , h ) A( c , h )  1]
PB ( a , b )  PB ( a , c )    ( h ) dh [ A( a , h ) A(b , h )  A( a , h ) A( c , h )]
PB ( a , b )  PB ( a , c )    ( h ) dh [1  A( b , h ) B ( c , h )]  1  PB ( b , c )
PB ( a , b )    ( h )dh A( a , h ) B (b , h ),
  (h )dh  1
Since A( a )   B ( a ) and PB ( a , a )  1, Bell wrote
PB ( a , b )     ( h ) dh A( a , h ) A( b , h )
Simultaneous definite values for quantum mechanically non-commuting
observables!
Since  A ( z )   B ( z ) and P ( z A , z B )  1, we write
P ( A ( x ),  B ( z ))    ( h ) dh A ( x )  B ( z )     ( h ) dh A ( x )  A ( z )!
CSU, Proc. SPIE Photonics 2007
If nonlocal influence are allowed then any classical theory (of the coin tossing
type) can be made to reproduce whatever correlations one demands!
Hence the strict logical implication of the experimental results is that a
classical theory of the type Bell considered can be a valid theory of
microscopic phenomena IF one allows nonlocality as an additional feature.
This then takes away the uniqueness of quantum theory, contrary to the
common belief.
1)
Correlation functions of quantum mechanics are direct consequence of the
CLASSICAL conservation laws arising in space-time symmetries (fundamental
conservation laws), applied to ensembles.
2)
Any theory that has a correlation function different from the ones in QM is
incompatible with the fundamental conservation laws and space-time symmetries,
and therefore it is unphysical. Local hidden variable theories fall in this class.
Bell’s inequalities can be obeyed (in the general case) only by violating a
fundamental conservation law, making them redundant in physics.
3)
The origin of Bell’s inequalities can be traced unambiguously to the single step of
ignoring wave-particle duality and has nothing to do with the violation of Einstein
locality .
4)
The logical implication of the experimental result that Bell’s inequalities are
violated is that a classical statistical theory can reproduce quantum correlations
(or any arbitrary correlation for that matter!) only if it violates Einstein locality, and
NOT that QM is nonlocal!
CSU, Europhys. Lett, 2006, Pramana-J.Phys (2006)
6. The belief of the reality of zero-point modes in vacuum
Energy of vacuum modes (Casimir Force)
En   n  12   where n  0,1,2,3...
E
n

d
No long-wave
cut-off outside
n
2 c
0.013
2
Pc (d )  


dyne
.
cm
240d 4
d 4
However, this accepted view clashes severely with cosmology
Cosmology:
General features that are observationally supported:
• The Universe is expanding at the rate of 2x10-18 m/s/m
• The density of the Universe is about 2x10-29 g/cm3
• There is a background radiation with the black body spectrum at the
temperature of 2.728 K
• The Universe is approximately isotropic and homogeneous at very large
scales
2
 
R  8 G 
2

H 

R
3
 
Rate of expansion is proportional to the
sqrt. of average density
Energy density of QUANTUM VACUUM
Energy density

1
3
1
  E /V 
h



;



d

2
i

0
V All Modes i
c
Even with a sharp cut-off in frequency (x-rays…)
> 1 g/cm3
But, observations (cosmology, astrophysics):
Average cosmic density < 10–29 g/cm3
COSMOLOGICAL CONSTANT PROBLEM
Something is seriously wrong with Quantum Field Theory
7. Black holes and information loss – serious misunderstanding of black hole gravity
2GM
Horizon Surface :  2  R
c
The entire problem is formulated with us or asymptotic frames as the observers
Questions:
1. Can we hide information inside a blackhole by throwing it in from far away?
2. Is there information loss?
A very important result from general relativity:
There is exponentially increasing redshift on all processes approaching a black
hole horizon by free fall.
Clearly, the observer who is not freely falling with whatever that is falling in
will never manage to put anything into the blackhole and will never face the
loss of information into the horizon. All free falls towards a blackhole as
observable from a frame far away is asymptotic in time and it needs infinite
time to reach the event horizon.
Entanglement = Nonlocality
This can be held only by somebody who is sinning hard and confuses
quantum description and a classical statistical description of
microscopic phenomena. If you describe phenomen involving
entnaglement using a classical statistical theory, then you need
nonlocality (influence outside the lightcone).