Maxwell equations from spin-1 transformations under boosts

[hellfire]

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.

 

[member 11137]

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)!

 

[hellfire]

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) .

 

[Norman]

There is weinberg's paper. You will have to pay for this paper or find a library with a subscription.

http://prola.aps.org/abstract/PR/v133/i5B/pB1318_1

Good luck.

 

[member 11137]

... However, the derivation of a similar equation for spin-1 systems is not given... I would appreciate any comments or any discussion about it

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 !

 

[hellfire]

Thanks again for the comments.

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 !

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.

 

[samalkhaiat]

maxwell equations from spin-1

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

:
OK, I looked at your home work and noticed the followings;
1) you said that the (R & L) phies are 2-component spinors, even though you used them everywhere as 3-comp. vectors.
2) at some stage, you defined 6 equtions! Any thing before & after that stage is irrelevant because, as I will show, those 6 equations are suficient to derive Maxwell Eq's in the limit m-->0;
Let us write your Eq's (22) and (23) in more general form,
(E + S.p)L = mA,
(E + S.p)R =-mA
and
p.L = mD,
p.R =-mD,
where m is mass and (A,D) are any quantities with right dimensions.
Have you checked the dimensions in your Eq's (22),(23)?
Now add and subtract my 2 pair, then use your Eq's(24) which I write them as
L = e + ib, R = e - ib,
you will get from the 1st pair;
Ee + i(S.p)b = 0,
iEb +i(S.p)e =mA.
and from the 2nd pair;
p.e = 0,
ip.b=mD.
Now integrate the above equations with EXP(ip.r), then and only then you can use (E,p)-->their quantum operator.
for example;
Integral[p.e(p)exp(ip.r)dp]=p^Integ[e(p)exp(ip.r)dp]=p^e(x)
To get Maxwell Eq's, just take the limit m-->0.
All this means that your defining Eq's(22,23,24) are indeed Maxwell Eq's in momentum space.
3) you "derived" Eq's (25) & (26) which give the electric (e) & magnetic (b) fields in terms of potentials (A,phi). Now these Eq's are gauge invariant, but most of your Equations,[28 & 30] for example, are not gauge invariant, this is odd.
see the following for spinor field formulations of Maxwell Eq's;
1 M.SACHS, J.MATH.PHYS. 3, 5(1962), 843-848.
2 H.MOSES, PHYS.REV. 113, 6(1959), 1670-1679.
3 M.SACHS, INTER.J.THEO.PHYS. VOL4, No2(1971), 145-157.
4 J GALLOP,====================, No3(1971), 185-188.
5 PAULI, & VILLARS, REV.MOD.PHYS, 21(1949), 434.
regards
sam

 

[hellfire]

Thank you samalkhaiat, this is a very carefully considered response.

1) you said that the (R & L) phies are 2-component spinors, even though you used them everywhere as 3-comp. vectors.

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.

2) at some stage, you defined 6 equtions! Any thing before & after that stage is irrelevant because, as I will show, those 6 equations are suficient to derive Maxwell Eq's in the limit m-->0;

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.

Now integrate the above equations with EXP(ip.r), then and only then you can use (E,p)-->their quantum operator.
for example;
Integral[p.e(p)exp(ip.r)dp]=p^Integ[e(p)exp(ip.r)dp]=p^e(x)
To get Maxwell Eq's, just take the limit m-->0.

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?

3) you "derived" Eq's (25) & (26) which give the electric (e) & magnetic (b) fields in terms of potentials (A,phi). Now these Eq's are gauge invariant, but most of your Equations,[28 & 30] for example, are not gauge invariant, this is odd.

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)?

 

[samalkhaiat]

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

.

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.

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?

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

 

[hellfire]

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.

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...

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.

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.

 

[samalkhaiat]

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

...
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].

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)

.
Yes you can do the same thing if you change you variables from (p) to (x).
regards
sam