Matej Pavsic
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        Answers to Some Valuable Comments
  • Rewriting the conventional constrained point-particle action in terms of polyvectors and keeping only a minimum of nontrivial polyvector components yields the unconstrained, Stueckelberg, action. This result is central to the author's philosophy, because an analogous relation also holds for membranes.

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    True. Notice that this also refers to rewriting the conventional field theory in terms of polyvectors, that is, in reformulating it in Clifford space (C-space).
    So (micro)causality, etc., is required to hold in C-space; this, according to my proposal, is now the true space in which physics is to be formulated. This is described in detail in my papers

 

 
  • Most physicists will not follow the author when he approves of the sacrifice  of (micro)causality and positivity of the energy in order to adhere to the Stueckelberg particle (which admits tachyons) and its second quantization or to the pseudo-Euclidean oscillator. (The main objection to unbounded negative energy is not gravitational repulsion but the loss of stability of our world!)

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    From the  papers and my book (p. 38)  it is clear that even in my formulation of the second quantization of the Stueckelberg particle the energy  (i.e.,  )  is  positive

    Pseudo-Euclidean oscillator is a different thing. In pages 96-99  of my book  and in
    a paper I discuss a toy model which is then used in the field theory described by the action (3.33). Roughly speaking, the field  has positive energy, whilst the field  has negative energy. And yet the system is stable, because not only the potential, but also the kinetic term for  is negative;  therefore the equations of motion for both fields are the same as if there were no change of sign  in front of the  terms in the action (3.33). This is illustrated in eqs.(3.2) of the toy model. The criterion for stability of such system is not the minimum of the overall potential energy , but the requirement that there is minimum in the-plane and maximum in the -plane. As far as gravitation is not included into the game, the system described by eq.(3.33) (the same for (3.1 )) is equivalent to the analogous system in Euclidean space.

(Micro)causality is supposed to hold in C-space.  Spacetime is a subspace of C-space.  Quantum field theory should be formulated in C-space along the well established rules.

 
 
The sign of the kinetic term is also changed. Therefore the stability for negative signature components is assured by the existence of the upper (and not lower) bound of the potential. The overall criterion for stability involves the metric and reads 

Examples of (numerical) solutions to the system of two coupled pseudo-Euclidan oscillators are given here (Ps  or nb)   and here (Ps  or nb )).


 
  • The expectation that an ultrahyperbolic metric signature will make string theory well-defined in any dimension is also unfounded.

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    That an ultrahyperbolic metric signature will make string theory well-defined in any dimension was indicated in a paper by Jackiw. He is quite right, and he wrote me a letter in which he expressed his agreement with my approach. One only needs to look a little bit outside of what we have learnt about negative energies and stability and reconsider the subject from the start within the new set of postulates.

 

 
 
  • There is complete lack of any reference to any major string theory papers. While this is certainly not a crime in itself it reinforces the feeling that the book has been written in a vacuum, unaffected by the developments in string and higher dimensional theories.

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    This remark refers to my book The Landscape of Theoretical Physics: A Global View; From Point Particles to the Brane World.  Since there about 250 references the book certainly has not been written in a vacuum. However the book is indeed not string oriented. There are references to the papers on branes, and regarding strings, there are some references to basic works and textbooks. The  underlying assumption in most modern approaches to string theory is the requirement that extra dimensions have to be compactified. Namely, strings and the accompanying extended objects (branes) live in spacetime which has necessarily more than 4 dimensions. Then there arises a problem of how to compactify all those extra dimensions. In my papers (which appeared much before the explosion of the "brane world" scenarios) and in my book I preferred to consider an alternative view, namely, that a 3-brane (sweeping  a 4-dimensional world manifold) already represents our observable spacetime; no compactification of dimensions of the embedding (bulk) space is needed.
     

 

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