[time 883] Re: Reviving particle democracy

Stephen Paul King (stephenk1@home.com)
Mon, 04 Oct 1999 23:50:33 GMT

Dear Mike,

        Perhaps Hitoshi Kitada's Local Systems theory might answer
some of the time questions: http://www.kitada.com/

Kindest regards,



On 2 Oct 1999 17:43:04 -0700, you wrote:

>The recent discussion on Fock space has prompted me to put forward a few
>speculative ideas I have been working on, but now in the context of Fock
>space. These ideas are very speculative and maybe completely crazy and
>are not ready for publication. So far I haven't seen any reason to think
>they are off beam, but then I am not very well-educated in the realm of
>field theory, where I expect the biggest challenge to come from. (I was
>active as a researcher during the early 70s in the field of S-matrix
>phenomenology and very attracted to the bootstrap hypothesis of particle
>democracy.) Since the ideas are very challenging to what I believe to be
>the conventional viewpoint, I look forward to sparking a lively debate
>which I hope will inform my own enquiries, if no one else's.
>Typically, one looks at the Hilbert space H_i for a specific single
>particle i, defined by a fixed mass, charge, etc. When taking the
>product Hilbert space H_1*H_2 one usually does it for two such specific
>particles which may or may not be identical. And so on for 3, 4, ... , n
>specific particles.
>My idea is to take the Hilbert space defined by the complete range of
>quantum numbers that can be attributed to any isolated physical system.
>This could mean not just the space of known single particles but it
>could also mean all possible composite systems of any number of
>particles and any hypothetical single particles that might exist. We are
>then no longer limited to any mass shell or even to a set of mass
>shells. The energy-momentum vector can range anywhere in 4-space allowed
>for a composite physical state. Isospin, strangeness and any other
>quantum numbers we might know about or even hypothesize about would also
>be allowed to range over all values allowed for a composite physical
>state instead of being limited to those of a particular particle. Note
>that we concern ourselves at this stage only with the total quantum
>numbers of the system and do not enquire about the internal structure.
>For this reason we shall call the system "closed".
>The state vector |S> for a given physical system S can therefore be
>anywhere in this space, and the probability of finding the system in a
>specific state |p> which is a basis vector of H (and might be a
>particular particle for instance) is |<p|S>|^2.
>Now suppose we have two or more such isolated systems. The Fock space
>can be built by symmetrizing the space H*H and so on. Note that we have
>dropped the subscripts 1 and 2 which formerly distinguished two specific
>particles because, now that H ranges over all possible physical states
>for each system and each system has the same range of possible physical
>states we cannot distinguish between state "1" and state "2".
>Furthermore, as described in my spin-statistics paper
>(http://xxx.lanl.gov/abs/quant-ph/9908078) it is possible to do this
>symmetrization for all possible spins whether fermionic or bosonic. (In
>what follows, when we write H*H, we intend that this space has been
>Now note that this new two-system system can also itself be described as
>a single isolated system if we don't enquire as to the separate systems.
>In other words we can either view the complete system as a single closed
>isolated system or as two isolated closed systems. In either case, it is
>apparent that the state vector describing the system can be placed in
>either H or in symmetrized H*H SIMULTANEOUSLY. Hence we can compute the
>probability that S is in a single state p from <p|S> or in a double
>state p_a,p_b from <p_a,p_b|S>. In general H, H*H, H*H*H,..etc all
>describe different ways of looking at the same physical space. This is,
>of course, nothing but a mathematical statement of the bootstrap
>principle of particle democracy. Every observable single particle is
>described by a vector in H which is also a vector in H*H (and therefore
>possibly describable as a two-particle system) and in H*H*H (a
>three-particle system) and so on.
>Now suppose we make two measurements. In the first case we detect a
>single set of quantum numbers p and in the second we detect a pair of
>sets of quantum numbers p_a and p_b. What has happened? Well we have
>observed a transition from p to p_a+p_b!! What is the probability of
>this transition? Well it is clearly |<p_a,p_b|p>|^2. In other words, the
>S-matrix element describing the transition from p to p_a+p_b is given by
>the scalar product of the two state vectors describing the initial and
>final states.
>In other words, the scattering operator is the identity operator! There
>is no interaction Hamiltonian -- merely spontaneous transition. What is
>more, the scalar product describing the transition amplitude should be
>familiar from a well-known problem in quantum mechanics -- the addition
>of discrete quantum numbers. For instance, if the only measurable
>quantum number in the universe was isospin, then our transition
>amplitudes would be just the Clebsch-Gordon coefficients relating two
>isospin states to a single composite isospin state and, as is well
>known, these are computed by reducing composite group representations to
>irreducible representations -- the direct analogue of our requirement
>that H, H*H and H*H*H... represent the same physical space!
>Before the howls of outrage get too loud, let me just anticipate some of
>the issues I expect to be raised...
>First objection: This model has (so far) no place for time. As we all
>know the decay p->p_a+p_b is described by a half-life or decay RATE.
>There appears to be nothing in the scalar product <p_a,p_b|p> which
>depends on time. Response: Actually this is reasonable. The probability
>of spontaneous decay is independent of how long the particle has been in
>existence. So how do we get a time scale into the picture? The answer is
>very simple, we compute ratios of different decay amplitudes. These give
>us ratios of decay rates, known as branching ratios. Any one branch can
>serve as our time scale to measure other branches. The absence of time
>in our model goes hand-in-hand with the absence of a Hamiltonian. We
>simply don't care about it, becaase it comes out in the wash anyway!
>Second objection: If particles are spontaneously transforming how do we
>ever get any continuity of observation? If we observe a particle p at
>one observation, don't we expect to observe it again a moment later?
>Response: Yes we do. The scalar product <p|p> is always 1. Every other
>scalar product <x|p> will be < 1. The continued existence of an observed
>state exactly as it is will always be the most likely result of a
>subsequent observation until a change occurs, and then the new state
>will be the most likely result of future observations. However, change
>is the most predictable feature of our universe. It is why we have any
>concept of time at all. Every observable rate of change can be reduced
>to a ratio of scalar products of the observed change relative to some
>standard scalar product describing what is essentially the behavior of a
>Third objection: if the scattering operator is the identity operator,
>doesn't this diagonalize the S-matrix, preventing any transition at all?
>Response: For the traditional S-matrix, defined by a mathematical trick
>(taking the limit t->infinity of asymptotically free particle states)
>this would be the case, because the initial and final state vectors are
>treated as part of the same orthogonal set and therefore all mutually
>orthogonal. (For this reason, transition can only take place, in the old
>model, when the hand of God intervenes in the form of an interaction
>Hamiltonian.) In my model described above, this is not the case, in
>general. Of course, there are specific examples where diagonalization is
>true. For instance if we only observe composite quantum numbers of
>single systems, then yes, the S-matrix will be diagonalized. This
>amounts to nothing more than conservation of the observed quantum
>numbers. But as soon as we permit simultaneous observation of two or
>more subsystems, then we break the condition that all state vectors are
>mutually orthogonal and get non-vanishing scalar products for differing
>Fourth objection: If the scattering amplitude is simply a computable
>scalar product, then why can't you compute it for all known possible
>values of observable quantum numbers, find the singularities and
>therefore explain the entire particle spectrum? Answer: Although simple
>coupling coefficients for discrete quantum numbers such as
>Clebsch-Gordon coefficients are computable with relative ease (assuming
>one knows the complete set of observable quantum numbers and the
>appropriate groups), I don't have enough knowledge myself to compute
>coupling coefficients for continuous variables such as energy-momentum. I
>am guessing that the first place to look for this would be the Poincare
>group, or some variation of it, but I don't know how myself. Any help
>with solving this problem would be much appreciated!!!

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