[ExI] QT and SR (was Re: Probability is "subjectivelyobjective".)

scerir scerir at libero.it
Wed Jul 16 17:08:57 UTC 2008


Jef writes:
> Your statement about algebraic contra geometric "nature" is
> intriguing.  After a bit of googling it's still not clear to me what
> you mean in any deep sense, and to speak of "nature" implies a very
> deep sense. [...]
> Thinking further, this may be yet another example of the
> subjective/objective, epistemic/ontic confusion which pervades much of
> this discussion. Does it really make sense to speak of algebraic
> contra geometric "nature" rather than algebraic contra geometric
> representations or operations?  Am I perhaps missing something
> fundamental here?

In QM if there are two systems, A in state |psi> and
B in state |phi>, the global system is described by
the state-vector |psi>|phi>.

But we may ask the opposite - and interesting - question.

Given a (general) state-vector of a global system, 
can one always write it as a direct product of the states 
separately describing each of the systems composing
the global system? 

The answer is negative.

Let me drop the formal proof. For a short one ...  
if |global> = SUM_ij c_ij |phi_i>|psi_j>,
where c_ij are complex coefficients,
with SUM_ij |c_ij|^2 = 1,
there is no way - if the superposition principle
is to be of general validity - to restrict the 
coefficients c_ij to have a factorization condition 
like the following c_ij = m_i n_j.

So there are state-vectors for which the above
factorization condition does not hold.

Given the well-known mantra that the state-vector
is the only link between the QM formalism and
the physical reality (whatever it means), we can 
say that, in many cases, QM does not ascribe
any separate reality (whatever it means) to the
systems whose combination (or composition) is
described by a state-vector for which the 
factorization condition does not hold. 

And this is the *algebraic* essence of the EPR
paradox. 

Of great EPR-interest is the spin singlet state,
given by |u+v->-|u-v+>. It is interesting to
point out that:
- it predicts opposite results for measurements
  of the third component of the spins of particles
  A and B,
- it predicts the result zero for the measurement
  of the total squared spin of particles A, B,
- it is rotationally invariant,
- it is not factorizable (we cannot describe separately 
  the entangled states or - according to the spirit of
  Copenhagen - we cannot ascribe separate reality to 
  each state).

Now the weird algebraic nature of non-separability
as Schroedinger (1935) becomes even weirder ....  
"When two systems of which we know the states by their
respective representations enter into a temporary
physical interaction due to known forces between
them and then, after a time of mutual influence,
the systems separate again, then they can no longer
be described in the same way as before viz. by endowing
each of them with a representative state vector.
I would not call that "one" but rather "the" 
characteristics of quantum mechanics."

That is to say, the algebraic nonseparability
becomes the geometric nonseparability or - as people 
like to call it - the geometric nonlocality.

As for the epistemic/ontic dilemmas ... I do not know.
Sometimes I'm inclined to believe that QM is a sort 
of (algebraic) operating system, or an assembler if you
like. If this is true, no surprise that the quantum 
gravity is difficult to find. You cannot get, sic et
simpliciter, the fusion of the operating system and
the hardware (or stuff, or gravitating space-time) ...
unless you know which is the operating system and 
which is the stuff.
 



 



 









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