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Maxwell equations from spin-1 transformations under boosts

  1. Oct 10, 2005 #1

    hellfire

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    The Dirac equation can be derived from the transformation properties of spin-1/2 systems under pure boosts. This derivation is presented Ryder's Quantum Field Theory. However, the derivation of a similar equation for spin-1 systems is not given. Following the same steps as in Ryder for the Dirac equation I have tried to work out the derivation of the equation that relates the two different parts of a two-component spin-1 system in chiral representation. I have also derived Maxwell’s equations from it. I am not sure whether the derivation is right, as it is not available in any of my references. I would appreciate any comments or any discussion about it:

    http://www.geocities.com/alschairn/diracmaxwell/diracmaxwell.htm

    It is a bit long, but most of the steps are very easy to follow.
     
  2. jcsd
  3. Oct 11, 2005 #2
    It's not too long; I had not the time to check it completely (mathematics) but the subject itself (I mean the idea contained in the way that was followed) looks interesting. It would be beautiful if we had such a way to get the equations of the relativity (spin 2)!
     
  4. Oct 11, 2005 #3

    hellfire

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    Thank you for your comment. I am actually very interested in any comment so I would like to encourage others to take a look. Also, I would like to point out here the two key points for the derivation: First, the choice of the 3x3 matrices as representations of the commutation relations of the Lorentz generators an, second, the definitions for the spinor components in terms of the electric and magnetic fields as well as the expressions for the potential. Almost all the rest is algebra as done in Ryder for the derivation of the Dirac equation. Whilst looking into some references I found out that there exists a paper by Weinberg in which it is derived a general equation relating the two components of a spinor for any spin. However, I was not able to find this paper in internet (see reference [4] in http://arxiv.org/abs/hep-th/9312090).
     
  5. Oct 11, 2005 #4
  6. Oct 13, 2005 #5
    Two remarks:
    1) Concerning equations (1) to (6b) of your demonstration for spin 1 particles, I wanted to notice that you can have a confirmation and an help in "Essential Mathematic Methods for Physicists; Weber and Arfken; International edition; Elsevier Academic Press; in English; 2004; Ch 3. pages 190-191 exercices 3.2.11. (it is a demonstration with other matrices than the generatord group) and 3.2.12 (It's your demonstration; you have done the exercice of this book)
    2) The article of Weinberg is 40 years old: just to note that we (poor amateurs) are running behind the train of the modernity without never being able to come in. It's normal. If it would be different, then we would be professionals !!!
     
  7. Oct 13, 2005 #6

    hellfire

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    Thanks again for the comments.
    Of course, I hope I did not cause the impression I was trying to do something new. I just made a “homework” and I am looking for discussion and comments to see whether I have made mistakes.
     
  8. Oct 15, 2005 #7

    samalkhaiat

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    maxwell equations from spin-1

     
  9. Oct 15, 2005 #8

    hellfire

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    Thank you samalkhaiat, this is a very carefully considered response.
    Each one is a component of a two-component spinor. Each of these components contains a 3-dimensional vector. It is similar to the Dirac case. Due to the structure of the Lorentz group you have always two-component spinors, corresponding to the representations of the two groups SU(2) x SU(2). The difference between the spin-1/2 and spin-1 case resides in the number of components of each of both components. In the Dirac case there are two components per component and in the spin-1 case there are three. I think that the fact that one can identify these three componets with the components of a 3-dimensional vector resides on the choice of the representation, taking the 3x3 rotation matrices S. You probably know this and I was not clear enough in my text; I will check this.
    Yes, you are right that one could take directly (22) (23) and (24) and then derive Maxwell equations. However, my goal was to start with the transformation properties under pure boosts. Do you think this was completely superfluous? Without (21) it seams to me that (22), (23) and (24) would be a guess, which, however, actually turned out to be justified and the correct one. But with (21) you can show that the definitions of A and the scalar potential [itex]\Phi[/itex] lead to the correct relations to E and B, prior to obtaining Maxwell's equations.
    I am not sure to understand this, but I will think about it... or may be you could elaborate a bit; why is it not allowed to directly substitute p with its corresponding quantum operator?
    I understand this; if there is a mass term, they will not be gauge invariant. But gauge invariance is recovered when m = 0. Do you think there is something wrong with (28) and (30)?
     
    Last edited: Oct 15, 2005
  10. Oct 15, 2005 #9

    samalkhaiat

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    .

    This what bother me; you have Eq(21) which is not gauge invariant leading to gauge invariant fields(e,b) satisfying gauge non-invariant Eq's (28) & (30).
    before putting m=0, the fields should not display gauge invariance. your costruction should lead to Proca's massive vector field. you should also show that m=0 reduces the number of degrees of freedom from 3 to just 2. Look at my posts in thread called spin-2 graviton.

    All your Eq's are in the mumentum space; L=L(p), e=e(p), etc. and p is a number, so to use p-->differential operator, you need to integrate your Eq's to get L(x), e(x),etc.Put it this way; you are deriving classical not quantum equations, so why do you use quantum operators.

    regards

    sam
     
    Last edited: Oct 15, 2005
  11. Oct 16, 2005 #10

    hellfire

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    OK, I will take a look. I am also not very happy with that equations; I am afraid that the separation I made between real and imaginary parts does not take into account that A, E and B could be also imaginary. But the whole thing becomes very dirty otherwise, and I was not sure...
    Yes, (21) is a classical equation. But, as far as I know, this is the usual way to proceed when deriving e.g. the Dirac equation. First you find out the classical equation and then substitute with the quantum operators (by the way this is the same procedure as done to derive the Schrödinger wave-equation). Ryder does not make this substitution for the Dirac equation and works always with the equation in momentum space, but Peskin and Schröder do; the equivalent to (21) for spin-1/2 particles is derived and, afterwards, p is substituted with the corresponding quantum operator. I am sorry, but I still do not understand what you mean.
     
    Last edited: Oct 16, 2005
  12. Oct 17, 2005 #11

    samalkhaiat

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    ...
    No this is not a problem, because (e,b,A & Phi) are real numbers in Maxwell theory.The problem that I see, even though I have not checked your method, is this;
    you put your Eq's(22,23,24) into Eq(21) and then you got the very same equations(22,23,24) out of (21) in Maxwell form. Well, I showed you, such substitution was irrelevant. Plus, somehow, you derived the normal differential relations between the fields [e(p), b(p)] and the potentials [A(p), Phi(p)] which is not right if you dot integrate out the momentum variable.
    Your method, if correct, should give you;
    [b(p) = ip X A(p)], instead of [b = curl A].
    .
    Yes you can do the same thing if you change you variables from (p) to (x).
    regards
    sam
     
    Last edited: Oct 17, 2005
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