**Stephen P. King** (*stephenk1@home.com*)

*Thu, 22 Apr 1999 12:58:16 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 256] Re: [time 255] Many Times and the computation of renormilazation"**Previous message:**Stephen P. King: "[time 254] Re: [time 253] Peter Wegner's paper"**Next in thread:**Hitoshi Kitada: "[time 256] Re: [time 255] Many Times and the computation of renormilazation"

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,...

(1988)

"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

P_a(t) =

e^(iHt)P_a(0)e^(-iHt)

(3.1)

(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) …

P_aN(t_N)] (3.2)

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.

(http://stkwww.fys.ruu.nl:8000/~ogcn/reclame/Ising.html)

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,

Stephen

**Next message:**Hitoshi Kitada: "[time 256] Re: [time 255] Many Times and the computation of renormilazation"**Previous message:**Stephen P. King: "[time 254] Re: [time 253] Peter Wegner's paper"**Next in thread:**Hitoshi Kitada: "[time 256] Re: [time 255] Many Times and the computation of renormilazation"

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