Talk:Klein–Gordon equation

Need history

When did they come up with this mod to Schrodingers equation??. A bit of history would be good here.--Light current 03:56, 2 October 2005 (UTC)

From memory, the equation came very shortly after Schrodinger's equation. It was the first attempt to form a relativistic wave equation. I am pretty sure it predated the Dirac equation. I think it was a pretty obvious extension of Schrodinger's equation to the relativistic case, which is why lots of people thought of it at the same time. --DannyWilde 04:15, 2 October 2005 (UTC)

Insufficient. Currently the Dirac equation page does a better job than this page, explaining the original physical motivation for its consideration as well as the reasons for the later abandonment of the K-G eq (as per a typical intro to RQM course). This page must furthermore explain how those reasons were nullified in order for the K-G eq to finally be readopted in modern QFT. E.g., potentially negative probability density does seem absurd, why isn't that a problem if describing a scalar field? 150.203.48.127 (talk) 02:52, 17 April 2008 (UTC)

Particles?

It seems inappropriate to say that the equation applies to particles in anyway, since the modern interpretation is that it is the equation satisfied by all quantum fields (not just scalars/pseudoscalars). Threepounds 02:49, 15 November 2005 (UTC)

Perhaps it would be useful to point out that while the Klein-Gordon equation yields the incorrect result for spin-1/2 particles (one needs the Dirac equation for that), it does yield the correct results for spin-0 particles such as pions.--Brennan 22:57, 27 January 2006 (UTC)

Does anyone has comments about the fact that Klein-Gordon equation is a second order equation? It seems that we can write first order equations only for half-integer spin particles...? What about photons and other spin 1 particles? Finally indeed there exist no spin 0 elementary particle discovered till now. Does this mean anything?. User:Dave.bradi 14:00, 15 November 2007 (UTC)

B. Roy Frieden's anonymous POV-pushing edits

B. Roy Frieden claims to have developed a "universal method" in physics, based upon Fisher information. He has written a book about this. Unfortunately, while Frieden's ideas initially appear interesting, his claimed method is conversial: Frieden's work is highly controversial; see for example

• Binder, Philippe M. (2000). "Physics from Fisher Information: A Unification (a review)". American Journal of Physics. 68: 1064–1065.
• Kibble, T. W. B. (1999). "Physics from Fisher Information: A Unification (a review)". Contemporary Physics. 40: 1999. (the reviewer has some positive comments but concludes that Frieden's work is "misguided")
• Case, James (2000). "An Unexpected Union---Physics and Fisher Information". SIAM News. July 17. eprint (highly favorable)
• Matthews, Robert (1999). "Physics and Fisher Information (a review)". New Scientist. January. unauthorized electronic reprint
• Physics from Fisher Information: A Unification (a review) from Cosma Shalizi (Computer Science, University of Michigan) (highly critical)
• Physics from Fisher Information (a review) from R. F. Streater (Mathematics, Kings College, London) (highly critical)
• Physics from Fisher Information thread from sci.physics.research, May 1999 (mostly critical)
• Fisher Information - Frieden unification Of Physics thread from sci.physics.research, October 1999 (mostly critical)

Note that Frieden is Prof. Em. of Optical Sciences at the University of Arizona. The data.optics.arizona.edu anon has used the following IPs to make a number of questionable edits:

1. 150.135.248.180 (talk · contribs)
   1. 20 May 2005 confesses to being Roy Frieden in real life
   2. 6 June 2006: adds cites of his papers to Extreme physical information
   3. 23 May 2006 adds uncritical description of his own work in Lagrangian and uncritically cites his own controversial book
   4. 22 October 2004 attributes the uncertainty principle to the Cramer-Rao inequality, which is potentially misleading
   5. 21 October 2004 adds uncritical mention of his controversial claim that the Maxwell-Boltzmann distribution can be obtained via his "method"
   6. 21 October 2004 adds uncritical mention of his controversial claim that the Klein-Gordon equation can be "derived" via his "method"
2. 150.135.248.126 (talk · contribs)
   1. 9 September 2004 adds uncritical description of his work to Fisher information
   2. 8 September 2004 adds uncritical description of his highly dubious claim that EPI is a general approach to physics to Physical information
   3. 16 August 2004 confesses IRL identity
   4. 13 August 2004 creates uncritical account of his work in new article, Extreme physical information
   5. 11 August 2004 creates his own wikibiostub, B Roy Frieden

These POV-pushing edits should be modified to more accurately describe the status of Frieden's work.---CH 23:54, 16 June 2006 (UTC)

I know nothing about Frieden's work, but it can be noted that the Fisher information metric is equivalent to the flat-space metric on an n-sphere, and when complexified is equivalent to the Bures metric of quantum mechanics. Well, the Fubini-Study metric I guess I forget the details. Misha Gromov wrote about this, nicely pointing out that if you take a bunch of ordinary statistical probabilities living on a simplex: $\sum _{n}p_{n}=1$ and take their square root you get quantum probability amplitudes: $\sum _{n}|\psi _{n}|^{2}=1$ which is the equation for a point on a sphere. Geometrical fun and games ensue, especially with $\sum _{n}p_{n}\log p_{n}$ where $p_{n}$ is a function of a parameter $\lambda$ and you consider $\partial p_{n}/\partial \lambda$ and presumably Frieden had fun with those games. I mean, so yes, you probably can derive QM from Fisher information, but so what? What does this tell you that you don't already know? 67.198.37.16 (talk) 17:45, 16 December 2020 (UTC)

Oh. Ewwww. I see what the problem is. This review: Studies in History and Philosophy of Modern Physics 33 (2002) 327–343 Essay review Physics fromFisher information D.A. Lavis, R.F. Streater. Indeed, there is cause for concern. 67.198.37.16 (talk) 18:15, 16 December 2020 (UTC)

Electromagnetic interaction

To Likebox: Thank you for correcting the rather stupid statement I put in when I created the section on Electromagnetic interaction. To avoid repeating my mistake, let me ask you which of the following is the correct action?:

${\mathcal {S}}=\int \left({\frac {1}{m}}\eta ^{\mu \nu }(-i\hbar \partial _{\mu }{\bar {\psi }}-eA_{\mu }{\bar {\psi }})(-i\hbar \partial _{\nu }\psi -eA_{\mu }\psi )-mc^{2}{\bar {\psi }}\psi \right)\mathrm {d} ^{4}x$

or

${\mathcal {S}}=\int \left(-{\frac {1}{m}}\eta ^{\mu \nu }{\overline {(-i\hbar \partial _{\mu }\psi -eA_{\mu }\psi )}}(-i\hbar \partial _{\nu }\psi -eA_{\mu }\psi )-mc^{2}{\bar {\psi }}\psi \right)\mathrm {d} ^{4}x$.

Notice that the net effect of the change is to change the sign of the A on the left. Thank you for your assistance. JRSpriggs (talk) 17:49, 29 June 2008 (UTC)

You chose a weird normalization, dividing the whole action by the mass, and it's a drag to carry around all those hbars and c's, and you put the "i"s in front of the derivative, where they would appear even when e=0, it is more consistent to put them in front of A. but anyway the action has the sign flip:

S= \int_x (\partial_\mu + i e A_\mu ) \phi^* (\partial_\mu - i e A_\mu)\phi

It has to to make the action real, the first factor has to be the complex conjugate of the second.Likebox (talk) 05:10, 30 June 2008 (UTC)

Thank you for your response. I am glad to see that it makes the action real.

As to the reason why I chose the normalization: I want all the actions (of various fields, particles, and such) to be compatible with each other to facilitate merging them together when looking at a composite system. Also, since the non-gravitational stress-energy tensor at Einstein–Hilbert action is defined in terms of a derivative of the non-gravitational action, I need to have the correct proportions. Otherwise, the results of practical calculations may be off by constant factors. As a standard, I chose to fix the coefficient of the non-relativistic kinetic energy. So the non-relativistic action of a single free test particle is

$S=\int {\frac {mv^{2}}{2}}dt$

from which we get that the kinetic term in the relativistic action of a non-quantum particle is

$S=\int -mc^{2}{\frac {d\tau }{dt}}dt.$

I am trying to standardize all my formulas for action and stress-energy to be compatible with that. In particular, to get the $T^{00}$ component of the stress-energy correct for a scalar particle, I considered a real-valued field which is a function only of time and chose a time when the first derivative is (instantaneously) zero. Under those conditions

$T^{00}=m{\bar {\psi }}\psi =-\eta ^{00}mc^{2}{\bar {\psi }}\psi$

should hold because ${\bar {\psi }}\psi$ should be the expected density of particles in quanta per cubic meter, and each quantum (presumably, being at rest) should have a relativistic mass equal to its rest mass, m. Notice that I use $\eta ^{00}={-1 \over {c^{2}}}$ because I am using SI units in which $\eta _{00}=-c^{2}$. JRSpriggs (talk) 14:13, 30 June 2008 (UTC)

The only circumstance that I can imagine when it would be (arguably) better to use SI units is if you are describing an engineering station which has human-scale girders and struts, human-time flywheels and pendulums, and a bunch of pions somehow directly interacting with them. SI units are just annoying: you have to use natural units to keep the formulas in your head.

I get that you want to add together all the actions, and make T the derivative of S with respect to the metric, those are noble causes. But if you divide by the mass, the m=0 limit is not smooth, and that's the important limit of short distances/highly relativistic particles.

The action \int |(\partial -i e A) \phi|^2 is fine, the normalization of the field is just different--- it's square doesn't give the density of particles. The quantity $m^{2}\phi ^{*}\phi$ is the energy density for a field of magnitude $\phi$, as you noted, but the field $\phi$ has a factor of $\scriptstyle 1/{\sqrt {E}}$ in it by convention when it acts as a creation operator, so that $|\phi |^{2}/m$ is the particle density not just $|\phi |^{2}$. That's the standard way people normalize relativistic fields. Your (nonstandard) convention is inconvenient because it uses the particle mass to define the field scale, but the small distance field fluctuations don't depend on the mass.Likebox (talk) 20:20, 30 June 2008 (UTC)

Hello. I don't mean to butt into this conversation but I think Likebox is correct in his analysis of why JRSpriggs' unusual convention has masses in a place that is... uncomfortable for people who do field theory on a daily basis. It seems like the issue is the claim that ${\bar {\psi }}\psi$ is the number of quanta per cubic meter. This is correct in a quantum-mechanical setup. However, strange as it may seem, this changes when we pass to quantum field theory, where particle number can change (via pair production, for example) and yet the energy ought to be conserved. Anyway, the unusual convention seriously encumbers the use of Wikipedia as a reference.Evanberkowitz (talk) 04:26, 25 August 2010 (UTC)

You mention $F^{\mu \nu }\partial _{\mu }\partial _{\nu }\phi \,$ in the article. I would like to point out that it is identically zero because F is antisymmetric and the derivatives commute. Even if the derivatives were covariant, they would commute since they are applied to a scalar and the Christoffel symbol is symmetric in its lower indices. JRSpriggs (talk) 18:18, 1 July 2008 (UTC)

Oops! Sorry. For a good nonrenormalizable term I should have used \partial_\mu (F_\mu\nu \partial_\mu \phi). Thanks for catching that.Likebox (talk) 18:36, 1 July 2008 (UTC)

Rats, that one is screwed up too. I found a better one and put it in the article.Likebox (talk) 18:40, 1 July 2008 (UTC)

Error in derivation of the Klein-Gordon equation

After taking a look at the derivation of the Klein-Gordon equation I think I may have come across an error in the derivation. In the portion of the equation where the energy operator is plugged in you have -i times hbar times the del operator. When this quantity is squared you should get hbar squared times the laplacian. Instead the derivation has -hbar squared times the laplacian. —Preceding unsigned comment added by 207.230.130.223 (talk) 18:46, 2 October 2008 (UTC)

$(-i)^{2}=-1\,$ JRSpriggs (talk) 05:54, 3 October 2008 (UTC)

Hydrogen atom solution

Has anyone thought to address the issue that the KG equation has a solution that predicts a charged-particle orbit with a binding energy of over 500 keV?Aqm2241 (talk) 06:54, 3 May 2011 (UTC)

Would you please give a link to the paper which made this prediction!

I do not see what the relevance of this is as the hydrogen atom is made of spin 1/2 particles, i.e. the proton and the electron, rather than scalar particles (spin 0) which are what the Klein–Gordon equation deals with. JRSpriggs (talk) 11:22, 3 May 2011 (UTC)

The relevance to me has to do with original research (that I can explain later if it might be productive), but other reasons are explained in H. N. Spector and J. Lee, “Relativistic one-dimensional hydrogen atom,” Am. J. Phys. 53 (3), p. 248, 3/1985 and, more recently, in J. Naudts, “On the hydrino state of the relativistic hydrogen atom,” http://arxiv.org/abs/physics/0507193. Aqm2241 (talk) 03:54, 4 May 2011 (UTC)

Thank you for the link to that interesting article. However, I am afraid that it is beyond my ability (at this time) to evaluate it. But let me ask you this — if there is a hydrino solution for the hydrogen atom with much lower energy than the previously known ground state, then why does hydrogen gas not collapse into that state en-mass? Since that would be an exothermic reaction, I would expect it to go forward as it should increase entropy. JRSpriggs (talk) 14:07, 4 May 2011 (UTC)

I wrote you a long response earlier, but it disappeared somewhere along the line. Basically, forbidden transitions (e.g., 0->0) take orders of magnitude longer. Also, high frequency drivers (e.g., a relativistic electron) cannot add much energy to an oscillator (the photon) with a much lower fundamental frequency. Thus, unless actually found, E = n h nu would be the only option, where n is an even integer >> 1 and the photons can add together in such a manner as to have net 0 angular momentum. Since E = n h nu is not = h n nu, this isn't even an option.

If you know an expert in the area, who might be interested in coauthoring a paper on the subject, I would appreciate a name.Aqm2241 (talk) 07:24, 11 May 2011 (UTC)

Masses without zero subscripts

Should there not be a lot of m₀s instead of ms around? what do you think?CecilWard (talk) 05:22, 2 January 2012 (UTC)

Why clutter things up with a lot of unnecessary subscripts? At this level, we are never going to talk about "relativistic mass" rather than "rest mass". Relativistic mass, if one had to refer to it, would be described as energy divided by the speed-of-light squared. JRSpriggs (talk) 12:45, 2 January 2012 (UTC)

Curious juxtaposition

Is it just me, or is the following juxtaposition in the lede expected to confuse:

It is the equation of motion of a [...] field whose quanta are spinless particles. [...] Any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, [...]

Any solution to the Dirac equation has spin 1/2, which is not a spinless particle. My interpretation is that the first statement only true if one (quite unnecessarily) imposes a spinlessness restriction on the field via definition (usually for purposes of simplicity of exposition). The second statement is true if this restriction is dropped, but only then. Is the Klein–Gordon equation considered in this article to apply to spinor fields or not? Let's clarify this. My own preference is to be encyclopaedic and not pedagogical, thus preferring the general case in which the Klein–Gordon equation is taken to apply to arbitrary particle fields (of any spin), making the second and not the first statement true. —Quondum 09:11, 7 June 2014 (UTC)

Perhaps the first sentence should be re-worded. I think you are misinterpreting it. It is not saying that the Klein-Gordon equation is only true for spin-zero particles. Rather it is saying that spin-zero particles merely obey it while particles of higher spin also obey other equations which imply the Klein-Gordon equation. JRSpriggs (talk) 10:02, 8 June 2014 (UTC)

I've reworded it to make this interpretation clearer. —Quondum 18:58, 8 June 2014 (UTC)

Current conservation

I added a section on the derivation of the current conservation for a complex scalar field. Someone whould also add a proof using the U(1) symmetry and Noether theorem (or add a link if this is already proved somewhere else). I'll do it myself when I have the time if no one else does that. Ggf4t (talk) 12:38, 13 January 2015 (UTC)

Undid revision [1]

We can't have statements like "...is obviously flawed..." when it comes to what the literature says (and is also true), especially not without a citation. The edit has some points, but needs a rewrite and a citation. I might give it a shot later if nobody gets there before me. YohanN7 (talk) 00:07, 28 March 2015 (UTC)

I don't know why you undid this. The flaw should be obvious to anyone with knowledge of field theory. Well good luck then.

Aoosten (talk) 22:06, 20 November 2015 (UTC)

I told you, it is because of the "obviously flawed" passage. The KG equation is more than a relativistic field equation for a scalar, but it doesn't make the interpretation flawed. Even if it was flawed, it couldn't be put into the article without a citation. YohanN7 (talk) 16:11, 23 November 2015 (UTC)

The text now states that "this form is interpreted as the relativistic field equation for spin-0 particles" (A) and that "any solution of the free Dirac equation is a solution of the Klein-Gordon equation as well" (B). These statements are incompatible. If you adhere to statement A you can not also accept B. Aoosten (talk) 17:52, 3 December 2017 (UTC)

To Aoosten: Notice the word "free" indicating the absence of interactions with other fields. If interactions are ignored, each component of a field of any spin will obey the Klein–Gordon equation. If interactions are considered, only the scalar and pseudo-scalar fields, i.e. fields with a single component, obey it. JRSpriggs (talk) 04:23, 9 December 2017 (UTC)

To JRSpriggs: Thanks, I noticed. In the presence of interaction you have to change the equation anyway.

Green's function and N≠4

A section on the retarded, advanced and Feynman propagators would be nice. The momentum-space form is obviously $1/(p^{2}-m^{2})$ and it's static, non-wave form fourier xform is the Yukawa potential and the non-static generic propagator has explicit forms given here and in several other places, e.g. a Feynman article from the 1950's on the theory of the positron. A discussion of the Lorentz invariance of the green's functions would be nice. A discussion of what happens in dimensions other than 4D is especially very interesting, since apparently, Huygens principle holds only in 3+1D or more generally, only for odd spatial dimensions. Google for "huygens hadamard" for more info. Hadamard conjecture. The results are very surprising and should be covered here. 67.198.37.16 (talk) 22:19, 15 January 2016 (UTC)

Lagrangian (action)

Maybe we should add the Lagrangian density (and add the Hamiltonian) with more common units:

${\mathcal {L}}={\frac {1}{2}}\hbar ^{2}\partial _{\mu }{\bar {\psi }}\partial ^{\mu }\psi -{\frac {1}{2}}m^{2}c^{2}{\bar {\psi }}\psi$

As taken from the Greiner of Field Quantization. This way the scalar field has dimension 1 (energy) in natural units. MaoGo (talk) 00:43, 4 December 2016 (UTC).

I think it is more common too. I can't recall having seen the one in the article. YohanN7 (talk) 16:18, 5 December 2016 (UTC)

The formula

${\mathcal {S}}=\int \left(-{\frac {\hbar ^{2}}{m}}\eta ^{\mu \nu }\partial _{\mu }{\bar {\psi }}\,\partial _{\nu }\psi -mc^{2}{\bar {\psi }}\psi \right)\mathrm {d} ^{4}x,$

is correct for action, S. This is just a matter of multiplying thru by powers of m and c to get the correct units.

The wave function, ψ, is a probability amplitude for a quantum to lie within a unit volume at the specified time and location. This means that

${\bar {\psi }}\psi$

is the probability density that a quantum will be found in that volume. So it must have units of meter ⁻³.

Action, S, must have units of kilogram·meter²·second ⁻¹. So its (Lagrangian) density must have units of kilogram·meter ⁻¹·second ⁻².

So the coefficient must have units of kilogram·meter²·second ⁻², that is, Joule.

A similar, but more complicated, argument works for the other term. JRSpriggs (talk) 03:51, 21 December 2019 (UTC)

Your argument is appealing but there are two critical issues. First, the wave function of a Klein-Gordon field is not interpreted as a probability amplitude, and no matter which unit the field is assigned to, the Noether charge always has an unitless integral. Second, this expression does not allow massless fields as m appears on the denominator. Leo-J-N-Jian (talk) 20:22, 23 May 2021 (UTC)

Metric signature

Can we change the metric signature to +--- ? This would make it compatible with standard textbooks and with other WP articles. Are there any QFT textbooks anywhere that use -+++ ? 67.198.37.16 (talk) 17:21, 16 December 2020 (UTC)

Non relativistic limit

I think the first time derivative of $\phi$ is approximated incorrectly. The term thrown away is a correction of size $E'/m$ relative to the one that is kept. In the second time derivative these corrections of this size are kept, so, if we want to truly expand the second time derivative to order $E'/m$ we should keep all terms of this size.

Another possible mistake is the sign of the overall phase in the definition of $\psi$ in terms of $\phi$, which should be reversed in my opinion to give $\psi =u(x)\cdot \exp {(-iE'/h)}\cdot \exp {(+im/h)}$ consitent with $E'={\sqrt {p^{2}+m^{2}}}-m$. — Preceding unsigned comment added by Foice (talk • contribs) 09:38, 26 June 2021 (UTC)

"Classical field"

What do they mean "Classical field" in the section "Non-relativistic limit" and then derive the Schrodinger Equation? How is Quantum Mechanics ever "classical"? I was expecting to see the case when h->0 and the correspondence to classical (which includes Relativistic) mechanics.

Symbols not appearing

Some of the symbols are just appearing as squares. I think it's delta squared but wasn't sure. 137.50.173.2 (talk) 11:41, 7 February 2024 (UTC)