Talk:Relativistic wave equations

Are they really quantum?

I'm not entirely happy that the first sentence talks about relativistic replacements for the Schrodinger equation. The given wave equations are usually the equations of motion for the underlying classical field, are they not? --Masud 08:35, 26 September 2006 (UTC)

Bargmann-Wigner equations for any-spin particles?

I think they are found here: [1]--Maschen (talk) 21:03, 15 January 2012 (UTC)

This has been notified here at the wikiproject talk page. There should be an article on this, but I don't understand these equations yet - else of course I would write about them...--Maschen (talk) 21:06, 20 January 2012 (UTC)

I plan to cobble together a few papers and start the article: [2][3][4][5][6][7][8][9] (this last one is not very helpful, but at least something...) Maschen (talk) 09:54, 17 August 2012 (UTC)

For the record (talking to myself of course) I created the article on 17/09/2012‎: Bargmann–Wigner equations. Maschen (talk) 17:08, 20 September 2012 (UTC)

Deleted "derivations" section

I deleted the section Derivation of basic relativistic quantum wave equations using 4-vectors since it hardly derives anything. It just uses four-vectors to construct relativistic wave equations, the bulk of the section mostly lists four vectors. A much better construction is to use representation theory of the Lorentz group. M∧Ŝc²ħεИ_τlk 15:00, 29 May 2016 (UTC)

Hi Maschen,

Could you place some of the construction of RWE's via representation theory of the Lorentz group on the page? I think it would be informative. As the page currently is, the RWE's are essentially just placed fully formed, without a clear reasoning behind the differences of each type or how they can be derived from more basic physical objects.

Thanks,

John Wilson

Sure, in time. I'll dig up an old paper or more (can't remember the names of the authors but they overview what we want). M∧Ŝc²ħεИ_τlk 18:27, 29 May 2016 (UTC)

Jaroszewicz and Kurzepa (1992) is the paper I meant (referenced in this article), Sexl and Urbantke (1992) is a great book also (not in this article). There is of course Weinberg's QFT (vol 1) and maybe Ryder's QFT (although it leans on the pedagogical side). I'll try in the next few days. M∧Ŝc²ħεИ_τlk 10:09, 31 May 2016 (UTC) amended M∧Ŝc²ħεИ_τlk 10:13, 31 May 2016 (UTC)

Actually, sorry, I don't think I can, best if someone who knows the subject do it. The relativistic transformation of a quantum state (under a representation of the Lorentz group) is easy enough to write down and reference, but I still don't (after years...) understand how to derive a relativistic wave equation for a given representation (i.e. explain why the Dirac equation has the form it does if you start from the D^(1/2,0)⊕D^(0,1/2) representation, and so on for higher spin equations). You can, of course, verify a given RWE is relativistically invariant, but this is not the same thing. At the very least, this article gives the simplest way to get to the KG and Dirac equations, using the energy momentum relation in the history section. M∧Ŝc²ħεИ_τlk 18:48, 4 June 2016 (UTC)

Derivation(construction) of basic relativistic quantum wave equations using 4-vectors

Start with the standard special relativity (SR) 4-vectors:

4-position $X^{\mu }=\mathbf {X} =(ct,{\vec {\mathbf {x} }})$

4-velocity $U^{\mu }=\mathbf {U} =\gamma (c,{\vec {\mathbf {u} }})$

4-momentum $P^{\mu }=\mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)$

4-wavevector $K^{\mu }=\mathbf {K} =\left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)$

4-gradient $\partial ^{\mu }=\mathbf {\partial } =\left({\frac {\partial _{t}}{c}},-{\vec {\mathbf {\nabla } }}\right)$

Note that each 4-vector is related to another by a Lorentz scalar:

$\mathbf {U} ={\frac {d}{d\tau }}\mathbf {X}$, where $\tau$ is the proper time

$\mathbf {P} =m_{o}\mathbf {U}$, where $m_{o}$ is the rest mass

$\mathbf {K} =(1/\hbar )\mathbf {P}$, which is the 4-vector version of the Planck-Einstein relation & the de Broglie matter wave relation

$\mathbf {\partial } =-i\mathbf {K}$, which is the 4-gradient version of complex-valued plane waves

Now, just apply the standard Lorentz scalar product rule to each one:

$\mathbf {U} \cdot \mathbf {U} =(c)^{2}$

$\mathbf {P} \cdot \mathbf {P} =(m_{o}c)^{2}$

$\mathbf {K} \cdot \mathbf {K} =\left({\frac {m_{o}c}{\hbar }}\right)^{2}$

$\mathbf {\partial } \cdot \mathbf {\partial } =\left({\frac {-im_{o}c}{\hbar }}\right)^{2}=-\left({\frac {m_{o}c}{\hbar }}\right)^{2}$

The last equation is a fundamental quantum relation.

When applied to a Lorentz scalar field $\psi$, one gets the Klein-Gordon equation, the most basic of the quantum relativistic wave equations.

$[\mathbf {\partial } \cdot \mathbf {\partial } +\left({\frac {m_{o}c}{\hbar }}\right)^{2}]\psi =0$: in 4-vector format

$[\partial _{\mu }\partial ^{\mu }+\left({\frac {m_{o}c}{\hbar }}\right)^{2}]\psi =0$: in tensor format

$[(\hbar \partial _{\mu }+im_{o}c)(\hbar \partial ^{\mu }-im_{o}c)]\psi =0$: in factored tensor format

The Schrödinger equation is the low-velocity limiting case (v<<c) of the Klein-Gordon equation.

When the relation is applied to a four-vector field $A^{\mu }$ instead of a Lorentz scalar field $\psi$, then one gets the Proca equation (in Lorenz gauge):

$[\mathbf {\partial } \cdot \mathbf {\partial } +\left({\frac {m_{o}c}{\hbar }}\right)^{2}]A^{\mu }=0$

If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation (in Lorenz gauge)

$[\mathbf {\partial } \cdot \mathbf {\partial } ]A^{\mu }=0$

More complicated forms and interactions can be derived by using the minimal coupling rule:

End of Derivation of basic relativistic quantum wave equations using 4-vectors

Hi Maschen, Since you decided not to put up the derivations based on " better constructions via representation theory of the Lorentz group", I thought it would be OK to put the 4-vector derivations on the Talk page so others could decide if it worth anything. It does actually get the KG and Schrödinger eqns, as well as the free Proca and Maxwell RWE's, as well as providing a starting point for the Dirac eqn.

208.104.19.227 (talk) 17:23, 3 August 2016 (UTC)

Yes, others should comment, but there is redundancy. In summary all you're using is relativistic invariance of the line element

$dX_{\mu }dX^{\mu }=(cd\tau )^{2}$

for the (+, −, −, −) metric, using the definition of 4-momentum

$P_{\mu }=m_{0}{\frac {dX_{\mu }}{d\tau }}$

for a massive particle, then replacing the 4-momentum by differential operators to obtain the KG eqn. Why include the wave 4-vector and why alternate between two notations (index and non-index)? Since a proper derivations section (not just a history section) is needed, a condensed version of what you wrote could be the first part of the new section, with a Lorentz group section following. M∧Ŝc²ħεИ_τlk 17:35, 4 August 2016 (UTC)

Hi Maschen,

You more or less just made my point. If one considers SR 4-Vectors as essentially fundamental entities, and then examines the very simple relations between them, RQM just falls right out...

Start with the standard special relativity (SR) 4-vectors:

4-position $X^{\mu }=\mathbf {X} =(ct,{\vec {\mathbf {x} }})$

4-velocity $U^{\mu }=\mathbf {U} =\gamma (c,{\vec {\mathbf {u} }})$

4-momentum $P^{\mu }=\mathbf {P} =\left({\frac {E}{c}},{\vec {\mathbf {p} }}\right)$

4-wavevector $K^{\mu }=\mathbf {K} =\left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)$

4-gradient $\partial ^{\mu }=\mathbf {\partial } =\left({\frac {\partial _{t}}{c}},-{\vec {\mathbf {\nabla } }}\right)$

All of these are standard SR physical 4-vectors, needing no axioms or assumptions from QM to justify using them. Each of them exists in purely SR terms.

Note that each 4-vector is related to another by a single Lorentz scalar:

$\mathbf {U} ={\frac {d}{d\tau }}\mathbf {X}$, where $\tau$ is the proper time

$\mathbf {P} =m_{o}\mathbf {U}$, where $m_{o}$ is the rest mass

$\mathbf {K} =(1/\hbar )\mathbf {P}$, which is the 4-vector version of the Planck-Einstein relation & the de Broglie matter wave relation

$\mathbf {\partial } =-i\mathbf {K}$, which is the 4-gradient version of complex-valued plane waves

Lorentz scalars also are an SR phenomena, not QM. Notice that everything after the 4-velocity is just a constant.

One can argue that the 4-Wavevector ($\hbar$) and 4-Gradient ($i$) are QM assumptions, but regardless, they are just simple constants. We are not assuming anything else that is normally considered as necessary for QM - no Uncertainty Principle, no Superposition Principle, no Operator Principle, no Born Probability, etc.

I include the 4-wavevector because it allows each relation to have a single constant between them, instead of jumping directly to ($i\hbar$). Also, it highlights the Einstein-Planck and de Broglie matter wave relations as being separate from the plane wave assumption. They really are different things and I feel that fact deserves attention.

So, with this simple construction, and the SR Lorentz Scalar Product rule, you get relativistic quantum wave equations with no other QM assumptions or axioms required... As you yourself just said: "In summary all you're using is relativistic invariance of the line element." Apparently, that is all that is required to get RQM.

With regard to redundancy, I am simply trying to emphasize a point in each block. The 1st block that there are a bunch of SR 4-vectors, and they are all pretty basic. The 2nd block is that these 4-vectors are all related to one another via single Lorentz Scalar constants. The 3rd block is that those constants consecutively build up to a basic quantum relation via the Lorentz Scalar Product rule. The final block is showing that many of the RWE's are just special cases of the basic rule we got this way.

The indexed versus non-indexed is just to show that it can be written either way, that 4-vectors are tensors, and thus frame invariant.

$\mathbf {\partial } \cdot \mathbf {X}$ is a 4-vector style, which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the 4-vector.

$\partial ^{\mu }\eta _{\mu \nu }X^{\nu }$ is a tensor index style, which is sometimes required in more complicated expressions, especially those involving tensors with more than one index, such as $F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }$.

208.104.19.227 (talk) 19:00, 4 August 2016 (UTC)

The wave 4-vector is irrelevant in the derivation of the equations, all that's needed is the QM operators for the energy and momentum, and the invariant spacetime interval (the energy-momentum relation is a consequence).

The fact that the De Broglie relations can be condensed into a simple proportionality between the 4-momentum and wave vector is a topic for the four vector and relativistic quantum mechanics articles and not needed in the derivation here. Also, setting up a neat list of 4-vectors and their relations with each other is also something very nice for the four vector article, but its off topic here.

If you want to start the section, feel free to go ahead and I will not remove. It's just my preference to present the shortest route to the solution without intermediate definitions and concepts. M∧Ŝc²ħεИ_τlk 19:34, 4 August 2016 (UTC)

Hi Maschen,

Ok, I put them back in the article. Thanks for talking about it. I would still like to see a construction/derivation in the manner that you suggested. I hope you will put one up in the future.

Thanks,

John Wilson 208.104.19.227 (talk) 19:43, 4 August 2016 (UTC)