**Hitoshi Kitada** (*hitoshi@kitada.com*)

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

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 257] Re: [time 255] Many Times and the computation of renormilazation"**Previous message:**Stephen P. King: "[time 255] Many Times and the computation of renormilazation"**Next in thread:**Stephen P. King: "[time 257] Re: [time 255] Many Times and the computation of renormilazation"

Dear Stephen,

A simple question.

----- Original Message -----

From: Stephen P. King <stephenk1@home.com>

To: <time@kitada.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
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*> 174- in Conceptual Problems in Quantum Gravity... Birkhauser. Boston,...
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*> (1988)
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*>
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*>
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*> "The fundamental formula of standard quantum mechanics gives the joint
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*> probability for the outcomes of a time sequence of "yes - no" questions.
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*> Such questions are represented in the Heisenberg picture by projection
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*> operators P_a(t) such that P_a^2 = P_a. The label a shows which question
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*> is asked, and the time at which it is asked. Questions asked at
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*> different times are connected by the Hamiltonian H through
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*>
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*> P_a(t) =
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*> e^(iHt)P_a(0)e^(-iHt)
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*> (3.1)
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*>
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*> (Throughout we use units in which \hbar = c = 1.) If a sequence of
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*> questions a_1 . a_N is asked at times t_1 </= t_2 </= . </= t_N, the
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*> joint probability for a series of "yes" answers is
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*>
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*> p(a_Nt_N, ., a_1t_1) = Tr [ P_aN (t_N) . P_a1(t_1)\pP_a1(t_1) .
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*> P_aN(t_N)] (3.2)
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*>
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*> where \p is the density matrix of the system and Tr denotes a trace over
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*> all variables. All the familiar features of quantum mechanics - state
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*> vectors, unitary evolution, the reduction of the wave-packet on an ideal
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*> measurement, and so forth - are summarized in the two formulae (3.1) and
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*> (3.2). Their utility as a compact and transparent expression of standard
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*> quantum mechanics has been stressed by many authors.
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*> This formula illustrates very clearly the special role played by time
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*> in quantum mechanics. First, the operators in (3.2) are *time ordered*.
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*> This is an expression of causality in quantum mechanics. Among all
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*> observables, time alone is singled out for this special role in
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*> organizing the predictive formalism. Second, it is assumed that *every*
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*> observation for which a prediction is made directly by (3.2) can be
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*> assigned a unique moment in time. This is a strong assumption. Unlike
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*> every other observable for which there are interfering alternatives
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*> (e.g., positions and momentum), this says that there is no observation
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*> that interferes with the determination of an observation's time of
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*> occurrence. We may, through inaccurate clocks or neglect of data, be
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*> ignorant of the precise time difference between two observations, but we
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*> assume that it could have been determined *exactly*. In such cases, we
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*> deal with ignorance as in every other case in quantum mechanics. We sum
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*> the *probabilities* over an assumed distribution of error to obtain the
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*> probabilities for the observation. We sum probabilities because we
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*> "*could* have determined the time difference but didn't." "
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*>
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*> First, would it make sense if we change (3.1) to:
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*>
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*> P_a(t)_LS_nl = e^(it_mH_(N-1) l) P_a(0) e^(-itmH_(N-1)l (I am not sure
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*> of how to write this correctly;) )?
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*>
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*> I am assuming that there are two possible "directions" of local time for
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*> each LS, since the movement of scattering particles does not necessarily
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*> have to be restricted to one direction, even if we can show that the
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*> propagator is noninvertible, e.g. satisfies f* exactness. Interestingly,
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*> this two-valuedness of the direction may be a good thing since we might
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*> be able to use the ideas contained in models of Ising spin.
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*> (http://stkwww.fys.ruu.nl:8000/~ogcn/reclame/Ising.html)
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*> Secondly, what would the form of (3.2) be? In the definition of
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*> Hitoshi's unitary group e^(-itmH_(N-1)l (t \elem R^1)on \H_nl, we find
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*> that the time is given with asymptotic accuracy in the limit of m -> ±
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*> oo (e.g., \infinity), this would imply that, working with Hartle's idea
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*> above, the assumption "we *could* have observed the time difference but
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*> didn't" is not correct.
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*> Third, given that "there are infinitely many times t = t(H_nl, \H_nl)
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*> each which is proper to the local system (H_nl, \H_nl)", should we not
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*> expect that where might be infinitely many time ordering of operators of
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*> the generalized form (3.2)?
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*> One final question: What is the disposition of the variable m? It is
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*> used to represent "mass",
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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"?
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*> There is a situation in renormalization that I find interesting in
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*> light of this last question. In Paul Teller's paper on pg. 74-89 of
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*> "Philosophical Foundations of Quantum Field theory" H. R. Brown & R.
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*> Harre eds. Claredon Press Oxford (1988), we find an interesting
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*> discussion of renormalization procedures. In particular on pg. 74 we
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*> find a discussion of the relationship between the mass of a particle and
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*> "self-interactions". As the number of self-interaction terms L increase
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*> the integral over them increases to infinity, if the "bare mass" m_0 is
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*> assumed to be infinite as well, the expression m_r = m_0 - I(L) gives us
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*> a finite "m_r" , but this involves a piece-wise finite process of
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*> computation using a finite "cut-off" for L and a proportionally finite
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*> m_L. ( I am trying to avoid the need to write out the equations
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*> explicitly ;) )
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*> Now, is the "m -> ± oo" term similarly piece-wise finite from the
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*> perspective of a given LS? Can we think of the situation of an evolution
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*> of interactions of LSs, from the external point of view, as being
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*> correlated with the computation of the piece-wise finite renormalization
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*> of a participating LS's center of mass? In other words, can we model the
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*> computation of the mass of a given center of mass particle as a function
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*> of the monotonic evolution of the interactions between a finite set of
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*> LSs as the number of LS -> \oo in a step-wise finite manner?
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*>
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*> Onward to the Unknown,
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*>
*

*> Stephen
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*>
*

Best wishes,

Hitoshi

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