Subject: Back to Basics

On Sun, 6 Sep 1998, Stanley Klein wrote:

> Hi Henry,
.....
> 
> Let me ask a more fundamental question. Why do you use the Heisenberg 
> picture with its emphasis on the operator S. 

The quantum state of a "pure state" physical system is sometimes
represented by the state VECTOR |PSI>, in Dirac notation, and sometimes
by the state (operator) S=|PSI><PSI|. The latter is preferred by rigorous
mathematical physicists for many reasons, a few of which are:
  
(1) The irrelevant overall phase is not brought in. 

(2) Everything is then expressed neatly in terms of operators, the 
    concept of `vector' not explicitly appearing. One needs only: 
    (a)  products of operators. [Operators are the quantum
         generalizations of the observable `numbers' of classical
         physic---the quantum effects arising because the propery AB = BA,
         valid for numbers, does not hold in general for operators, and
    (b)  the trace operation, which is the quantum version of integration
         over phase space with constant weight unity.    

(3) Statistical mixtures are representable with no need to change
    anything.

(4) States of subsystems can be defined: The state S_b of brain b is
    S_b= Tr(b~) S , where Tr(b~) means the trace over all degrees of
    freedom except for those of the brain b. 
 
(5) The reduction rule is S-->(P_e S P_e) := S_e, 
    where P_e eliminates components of S that are ruled out 
    by experience e.

(6) If P_e acts only on degrees of freedom associated with brain b, 
    then the reduction rule is equivalent to S_b-->(P_e S_b P_e).

(7) If e is a possible experience when the state of knowledge is  
    represented by S, then if the question put to nature is:
    "Will e occur now?", the probability that nature's answer will be
    YES is Tr S_e/Tr S.


> So the reduction event is   S 
> -> Pe S Pe. [Did I get that right?]

Yes.


> Wouldn't it be  simpler in communicating to nonphysicists to adopt the 
> Schroedinger  representation where states evolve? Then S -> Pe S.  
> This is somewhat simpler, and easier to picture. 

The issue is not Schroedinger versus Heisenberg picture, but whether
to stick with operators, and not bring in vectors. 

I am sure that it will be much easier for nonphysicists to understand 
the very simple idea that the `numbers' that represent the values of
observable quantities in classical physics must be replaced by operators,
which can be multiplied to give new operators, but for which order of 
the factors in a product  matters: simple examples with matrices will make  
this noncommutability clear. No need to confuse people with `vectors'.

The clincher is the other key formula, readily derived from the basic ones
given above, connecting the operator X corresponding to  some observable
feature of nature that corresponds to a number x [in the sense that any
measurement/observation of that feature will yield a numerical value x].
 
The formula is

<x> = Tr XS/TrS:

where <x> is the average value of the number x in the statistical ensemble
formed by summing over the possible values of x, each weighted by the
probability for nature to choose that value.


> Is the picture a little like many worlds except where 
> branches get pruned at every experience (so not it's not really many worlds 
> at all, but that picture might help).
> 
 
Basically yes! That picture gives the "normal" dynamics that I discussed.
Assume that the state evolves into a form that can be written as a  sum
of terms corresponding to mutually distinct possible experiences
of some observe: i.e., the state S(t) evolves into a form such that 

S_b = S_e1 + S_e2 + S_e3 + ... ,

where e1, e2, e3 ,... are mutually incompatible experiences, so that
for each N,

S_eN = P_eN S_b P_eN.

The mutual incompatibility of the different possible experiences means

P_eN P_eM = 0 for N different from M.

Thus S--> P_eN S P_eN will give

S--> S_eN:

the state S of the universe will be reduced to the part compatible with
experience eN.

> I know what would really help me is a concrete example using the double 
> slit and an observer. The observer would be having many experiences during 
> the time of the photon's travels in the closed box. Let |Oti> be the 
> observer's experience at time ti. I think I'll use the bra-ket notation for 
> clarity. Let P be the state of the photon. Then a movie of the world might 
> be:
> 
> |Ot1> |Pstart>
> ....
> |Ot20> |sum0(Pi)>  where Pi are locations between start and slits.
> ...
> |Ot50> |Pu + Pl>   where Pu and Pl are the upper and lower slits (I'm 
>                            ignoring the normalization factor.
> |Ot60> |sum(Pi)> where Pi are locations between slits and film.
> 
> |Ot90> |sum(Pi)> where Pi are locations on film
> 
> |Ot140>|P130> where P130 is the 130 spot on film
> 
> Incidentally included in the sums are lots of funny things like the photon 
> becoming an electron positron pair, and the photon hopping in funny places, 
> not going in a straight line. 
> 
> I am confident that I must have this picture wrong. The thing I probably got 
> right is the |Oti> which are the experiences of which the world is made. 


Experiences are REPRESENTED in Hilbert space.

If we adopt the Schroedinger picture of an evolving state of the universe,
S(t), then this state is defined at all t, and makes "jumps" when
experiences occur. I am not too comfortable with your statement

"Let |Oti> BE the observer's experience at time ti."

The observer's
experience IS the observer's experience. It comes in discrete units, and
each such experience e is represented in the evolving quantum state of
the universe S(t) by a jump at some time t_e. In your double-slit
example there may have been some earlier experience of observer O
[i.e., experiences represented by jumps in S_b(t), where b is the
body/brain of observer O] associated with his setting up this experiment,
but then, for the period that you are talking about, the observer is
effectively isolated from the experiment, and the experience represented
by the jumps in S_b are not particularly relevant to the course of events
in the experiment. You must pursue the description further: we could very
well talk about the film, and about taking the film to the photo-shop where
it is automatically developed, unseen by human eyes etc, until some months
later some grad student O looks at the film and finds (let us suppose
a high-tech experiment where only one photo was recorded on this film)
that the the photon landed on the grid in square #208. Or one could
imagine, instead, a more high-speed causal chain where there appears
promptly (within a millisecond) a displayed "#208". The graduate student
O has a brain that is able to support the experience that he describes to
himself and his colleagues as "#208 appeared", and he goes on to make a 
corresponding entry in his log book. If we were to follow the evolving
state S(t) of the universe during this period we would have all the things
happening that you mention. But the state S_b of the brain of O is
evolving independently of the progress  of the experiment until the causal
link to it causes the S_b to react: S_b will evolve into a sum,

S_b = P_e1 S_b P_e1 + ... + P_e208 S_b P_e208 + ... 

because the Schroedinger evolution generates a superposition of these
possibilities, and no reduction is supposed to have yet occurred.

It is supposed that the conditions are met NOW, and the set of
questions "Will experience e#N occur?" are all posed. [The order 
of posing does not matter, if P_e#N P_e#M =0 for all N different from M.] 

One experience, say e#208, will be picked by nature, and the state
of the universe will become S_e#208 = P_e#208 S P_e#208, which will
then evolve to later times t from that form at time t= t_e#208.

> But 
> how do I represent the photon's funny actions. There is an aspect of 
> Copenhagen that I'm not supposed to talk about P while it isn't part of the 
> experience. But in your unitary view S has everything in it. 

But the possible experiences under discussion here are
represented by possible jumps in the evolving state S_b(t) of the brain 
(or more generally the body) of observer O, who IS just the sequence
of experiences that are bound together by the dynamical connections
represented by the evolving S_b(t). This state of the brain S_b(t),
being defined as Tr(b~) S(t),  is  an aspect of the evolving state
of the universe S(t).

I hope this explicit description of the "basics" will make
the picture clearer. 

Henry