Stephen P. King (email@example.com)
Thu, 22 Apr 1999 12:58:16 -0400
Dear Hitoshi and friends,
A quote from "Time and Prediction in Quantum Cosmology" J. Hartle pg
174- in Conceptual Problems in Quantum Gravity... Birkhauser. Boston,...
"The fundamental formula of standard quantum mechanics gives the joint
probability for the outcomes of a time sequence of "yes - no" questions.
Such questions are represented in the Heisenberg picture by projection
operators P_a(t) such that P_a^2 = P_a. The label a shows which question
is asked, and the time at which it is asked. Questions asked at
different times are connected by the Hamiltonian H through
(Throughout we use units in which \hbar = c = 1.) If a sequence of
questions a_1 … a_N is asked at times t_1 </= t_2 </= … </= t_N, the
joint probability for a series of "yes" answers is
p(a_Nt_N, …, a_1t_1) = Tr [ P_aN (t_N) … P_a1(t_1)\pP_a1(t_1) …
where \p is the density matrix of the system and Tr denotes a trace over
all variables. All the familiar features of quantum mechanics - state
vectors, unitary evolution, the reduction of the wave-packet on an ideal
measurement, and so forth - are summarized in the two formulae (3.1) and
(3.2). Their utility as a compact and transparent expression of standard
quantum mechanics has been stressed by many authors.
This formula illustrates very clearly the special role played by time
in quantum mechanics. First, the operators in (3.2) are *time ordered*.
This is an expression of causality in quantum mechanics. Among all
observables, time alone is singled out for this special role in
organizing the predictive formalism. Second, it is assumed that *every*
observation for which a prediction is made directly by (3.2) can be
assigned a unique moment in time. This is a strong assumption. Unlike
every other observable for which there are interfering alternatives
(e.g., positions and momentum), this says that there is no observation
that interferes with the determination of an observation's time of
occurrence. We may, through inaccurate clocks or neglect of data, be
ignorant of the precise time difference between two observations, but we
assume that it could have been determined *exactly*. In such cases, we
deal with ignorance as in every other case in quantum mechanics. We sum
the *probabilities* over an assumed distribution of error to obtain the
probabilities for the observation. We sum probabilities because we
"*could* have determined the time difference but didn't." "
First, would it make sense if we change (3.1) to:
P_a(t)_LS_nl = e^(it_mH_(N-1) l) P_a(0) e^(-itmH_(N-1)l (I am not sure
of how to write this correctly;) )?
I am assuming that there are two possible "directions" of local time for
each LS, since the movement of scattering particles does not necessarily
have to be restricted to one direction, even if we can show that the
propagator is noninvertible, e.g. satisfies f* exactness. Interestingly,
this two-valuedness of the direction may be a good thing since we might
be able to use the ideas contained in models of Ising spin.
Secondly, what would the form of (3.2) be? In the definition of
Hitoshi's unitary group e^(-itmH_(N-1)l (t \elem R^1)on \H_nl, we find
that the time is given with asymptotic accuracy in the limit of m -> ±
oo (e.g., \infinity), this would imply that, working with Hartle's idea
above, the assumption "we *could* have observed the time difference but
didn't" is not correct.
Third, given that "there are infinitely many times t = t(H_nl, \H_nl)
each which is proper to the local system (H_nl, \H_nl)", should we not
expect that where might be infinitely many time ordering of operators of
the generalized form (3.2)?
One final question: What is the disposition of the variable m? It is
used to represent "mass", but is it an observable? When we say m -> ±
oo, do we mean "as mass increases to infinity"?
There is a situation in renormalization that I find interesting in
light of this last question. In Paul Teller's paper on pg. 74-89 of
"Philosophical Foundations of Quantum Field theory" H. R. Brown & R.
Harre eds. Claredon Press Oxford (1988), we find an interesting
discussion of renormalization procedures. In particular on pg. 74 we
find a discussion of the relationship between the mass of a particle and
"self-interactions". As the number of self-interaction terms L increase
the integral over them increases to infinity, if the "bare mass" m_0 is
assumed to be infinite as well, the expression m_r = m_0 - I(L) gives us
a finite "m_r" , but this involves a piece-wise finite process of
computation using a finite "cut-off" for L and a proportionally finite
m_L. ( I am trying to avoid the need to write out the equations
explicitly ;) )
Now, is the "m -> ± oo" term similarly piece-wise finite from the
perspective of a given LS? Can we think of the situation of an evolution
of interactions of LSs, from the external point of view, as being
correlated with the computation of the piece-wise finite renormalization
of a participating LS's center of mass? In other words, can we model the
computation of the mass of a given center of mass particle as a function
of the monotonic evolution of the interactions between a finite set of
LSs as the number of LS -> \oo in a step-wise finite manner?
Onward to the Unknown,
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