On Mon, 3 Jan 2011 01:38:35 -0800 (PST), Message-ID:
Subject: The Extended Relativity Theory In Clifford Spaces
RE: C. Castro and M. Pavsic, The Extended Relativity Theory
In Clifford Spaces, December 30, 2003, 56 pp.; file name: Carlosextended
scale relativity.01.pdf, 660,954 bytes
Dear Carlos and Matej,
Thank you for your very interesting paper. May I ask you for some clarifying comments, strictly private.
p. 3: "In this new physical theory the arena for physics is no longer the ordinary spacetime, but a more general manifold of Clifford algebra valued objects, noncommuting polyvectors."
p. 3: "In particular, the polyparticle dynamics in C-space, when reduced to 4-dimensional spacetime leads to the Stuckelberg formalism and the solution to the problem of time in Cosmology ."
***Matej: Ref.  is your book, please tell me about your understanding of the the problem of time in Cosmology, and the solution proposed. Mine is at
p. 8: "In MInkowski spacetime M_4 -- which is a subspace of C-space -- we observe the intersections of Clifford lines with M_4. And these intersections appear as localized extended objects, p-loops, described above."
***Please see the proper time [tau] in
E. Recami, Ruy H.A. Farias. A simple quantum equation
for Decoherence and dissipation (through interaction with the environment),
R.H.A. Farias, E. Recami. Introduction of a Quantum of
Time (chronon), and its Consequences for Quantum Mechanics,
R.H.A. Farias and E. Recami: "There are three equations, -- retarded, symmetric, and advanced Schrödinger equations, all of them transforming into the (same) continuous equation when the fundamental interval of time (that now can be called just tau) goes to zero."
p. 11: "In Stuckelberg theory the parameter of evolution is not the coordinate t=x^0. The evolution parameter is an extra parameter [tau] which is invariant under Lorentz transformations in M_4. For the extra parameter we can take one of the extra coordinates that enter our action (19). One natural choice for the evolution parameter is the scalar component [sigma] of a polyvector X (eq. 4)."
p. 11: "In nature indeed obeys the dynamics in Clifford space, then a particle, as observed from the 4-dimensional Minkowski space, can be accelerated beyond the speed of light , provided that its extra degrees of freedom x^.., x^..., ... , are changing simultaneously with the ordinary position x^.. . In C-space we thus retain all the nice features of relativity, but in the subspace M_4 we have an unconstrained Stuckelberg theory in which tachyons are not paradoxical and are consistent with the quantum field theory as well ."
p. 27: "The approach in refs. [16, 15, 17, 8] is different. We start from the usual formulation of quantum theory and extend it to C-space. We retain the imaginary unit i . Next step is to give a geometric interpretation to i . Instead of trying to give a geometric origin of i in *spacetime* we adopt the interpretation proposed in  according to which the i is the bivector of the 2-dimensional *phase space* (whose direct product with the n-dimensional configuration space gives the 2n-dimensional phase space). This appears to be a natural assumption due to the fact that complex valued quantum mechanical wave functions involve momenta p^m and coordinates x^m (e.g., a plane wave is given by exp(ip_mx^m), and arbitrary wave packet is a superposition of plane waves)."
***The last paragraph of p. 37 is very elucidating too.
***In summary, can you include the results by Erasmo Recami and Roy Farias regarding their [tau] , and develop an interpretation of QM that people like me might understand? To me the most difficult part from your review paper is the evolution parameter [tau] .
***The way I understand [tau] is that it refers to the proper "time" of a photon, hence no physical clock can read it,
K.S. Brown, Quantum Interactions on Null Surfaces,
***Am I wrong? I will appreciate the feedback from your colleagues too.
***Best regards - Dimi
Note: Talking about the unresolved issue of C-space tachyons (see above), see yet another mirror world in:
Andrey A. Grib, Nonzero cosmological constant and the many vacua world, gr-qc/0311048
"Tachyons nonobservable for visible
matter can be present in the dark matter, leading to some effects of nonlocality
in the space of the Universe.
I don't think that our "usual nervous mechanism" is the reason why we don't observe tachyons. Andrey Grib and I discussed this very same issue in his office on December 28, 1983. Twenty years later, he still talks about tachyons and "another world", and doesn't even mention the possibility that all these tachyons and "other worlds" could be an artifact from the treatment of time in present-day theoretical physics. The UNspeakable Holon does not live on the spacetime hypersurface, hence it is not, and cannot, be modeled with tachyons and mirror worlds. Back on December 28, 1983, I gave a simple example to Andrey Anatolievich for such potential reality from Russian literature (he knew very well the research by Prof. Vassiliy Nalimov from Moscow State University). See a widely known book by Rupert Sheldrake, and recall that we cannot find a relativistic description of the so-called collapse of the wave function since 1931.
There is no need to invent the wheel. In Russian: Nel'zia izobretat' velosipeda, Andrei Anatolievich.
You wrote: "We have also discussed why the volume [omega] of the spacetime filling brane plays the role of the global time mode, the Stuckelberg parameter  and its relation to the two modes of time given in ."
which can be interpreted (Kevin Brown) as 'interactions on null surfaces',
By 'local time mode' I understand the time read by your wristwatch: you observe things always post factum, in your past light cone. Hence the global time mode -- and the pre-geometrical "stage" of spacetime -- are hidden due to the speed of light: the spacetime "separation" between the transmission and absorption of a photon is zero, according to your (tardyon) clock, so you can never observe the pre-geometry of spacetime "between" two events, [A] and [B],
It seems to me that the global time mode is relevant to the Quantum of Time (chronon) introduced by Farias & Recami [Ref. 4], as I mentioned at
I will be very happy if you and Matej help me with your Clifford algebra.
Specifically, how is the global mode of time, as explained above, related to the Stückelberg-type parameter [Ref. 1]?
Is the the global mode of time *identical* to that extra, invariant, parameter which serves the role of evolution time [Ref. 3], hence providing a resolution of the problem of time in quantum gravity [Ref. 5]?
You can read this email also at
[Ref. 1] Carlos Castro. The programs of
the Extended Relativity in C-spaces: towards the physical foundations of
String Theory. Tue, 7 May 2002 19:23:48 GMT,
"What seems remarkable in this scheme of things is the
nature of the signatures and the emergence of two times. One of the latter
is the local mode, a clock, represented by t and the other
mode is a "global" one represented by the volume of the space-time filling
brane [omega]. For more details related to this Stuckelberg-type parameter
and the two modes of time in other branches of science see .
"Two times. Universal time arrow. Why universe may expand forever.
"We have also discussed why the volume [omega] of the
spacetime filling brane plays the role of the global time mode, the Stuckelberg
parameter  and its relation to the two modes of time given in .
It is possible that the fundamental constants, like the fine structure
constant, themselves depend on this global mode time parameter [omega].
As the universe expands, it does according to this universal time arrow.
Since this arrow points forward then one should expect that [omega] will
increase and not surprisingly the universe will expand forever."
[Ref. 2] Dimi Chakalov. Two modes of time:
[Ref. 3] Matej Pavsic. Clifford Algebra
Based Polydimensional Relativity and Relativistic Dynamics. Thu, 23 Nov
2000 20:15:53 GMT,
"Today the so called Fock-Schwinger proper time formalism is widely recognized for its elegance and usefulness, especially when considering quantum fields in curved spaces. There are two main interpretations of the formalism:
"(i) According to the first one, it is considered merely as a useful calculational tool, without a physical significance. Evolution in [tau] and the absence of the constraint is assumed to be fictitious and unphysical. In order to make contact with physics one has to get rid of [tau] in all considered expressions by integrating them over [tau] . By doing so one projects unphysical expressions into the physical ones, and in particular one projects unphysical states into the physical ones.
"(ii) According to the second interpretation, evolution in [tau] is genuine and physical. There is indeed the dynamics in spacetime. Mass is a constant of motion and not a fixed constant in the Lagrangian. Such an approach was proposed by Fock (1) and subsequently investigated by Stueckelberg (2), Feynman (3), Schwinger (4), Davidon (5), Horwitz (6), Fanchi (7) and many others (8,9).
"In this paper I am going to show that yet another, widely
investigated formalism based on Clifford algebra brings a strong argument
in favor of the interpretation (ii). Clifford numbers can be used to represent
vectors, multivectors and, in general, polyvectors (which are Clifford
aggregates). They form a very useful tool for geometry. The well known
equations of physics can be cast into elegant compact forms by using the
geometric calculus based on Clifford algebra.
"The extra variable s is related to the parameter
[x] through a choice of "gauge", that is by choice of the Lagrange multiplier
"The polyvector action and DeWitt-Rovelli material reference fluid
"In a remarkable paper (17) Rovelli considered in modern
language the famous Einstein "hole argument" which shows that points of
empty spacetime cannot be identified. For a precise formulation the reader
is adviced to have a look at Rovelli's paper. Here I present the argument,
as I understand it, in a simplified and compact way.
"If we wish to build up a theory in which spacetime points
could be identified, we have to fill spacetime with a *reference fluid*.
Such an idea was earlier considered by DeWitt (18), and revived by Rovelli
"Geometric calculus based on Clifford algebra leads to the point particle action with an extra variable -- the clock variable -- which enables evolution in spacetime. In other words, our model with the polyvector action allows for the dynamics in spacetime (relativistic dynamics). Relativistic dynamics is a necessary consequence of the existence of Clifford algebra as a general tool for description of geometry of spacetime.
"Moreover, when considering dynamics
of spacetime itself, such model, in my opinion, provides a natural resolution
of "the problem of time" in quantum gravity. A number of researchers have
come close to the viewpoint that even in gravity one has to introduce an
extra, invariant, parameter which serves the role of evolution time (19,20,21).
The latter parameter, as already stated, in the polyvector generalization
of physics is not postulated but is present automatically."
[Ref. 4] R.H.A. Farias, E. Recami. Introduction
of a Quantum of Time (chronon), and its Consequences for Quantum Mechanics.
"There are three equations, -- retarded, symmetric, and
advanced Schrödinger equations, all of them transforming into the
(same) continuous equation when the fundamental interval of time (that
now can be called just tau) goes to zero."
[Ref. 5] C.J. Isham and J. Butterfield.
On the Emergence of Time in Quantum Gravity.
Subject: Re: hep-th/0205065 and refs.
 and  therein
I'm afraid you have misunderstood the idea of two modes of time, as I tried to explain in my email to you from Wed, 08 May 2002 12:54:18 +0300,
Since you quoted my paper as ref. , please note that there is a typo in your hep-th/0205065: biocausality, not bicausality.
I believe you have developed a very interesting theory which deserves serious discussion, but I can't see how it could be related to my paper.
Perhaps it would be a good idea if you reconsider the need to give references to my paper, since I will not attend the NATO Workshop and will not have the chance to explain the idea of two modes of time,
I wish you and all your colleagues a pleasant and fruitful conference.