[time 256] Re: [time 255] Many Times and the computation of renormilazation

Fri, 23 Apr 1999 02:18:34 +0900

Dear Stephen,

A simple question.

----- Original Message -----
From: Stephen P. King <stephenk1@home.com>
Sent: Friday, April 23, 1999 1:58 AM
Subject: [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
> 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",

Here do you mean the same thing by this m and the m in t_m in the above
formula corresponding to (3.1)?

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
>

Best wishes,
Hitoshi

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