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Kaluza–Klein theory is theory which uses 5th dimension to unite Gravity with Electromagnetism.

does this theory have any short fall or is it perfect?

If someone can help me with original article it would be nice of them.

Regards,

Bhushan Poojary

In 1919, the mathematician Theodor Kaluza began to play around with Albert Einstein’s formulas for gravity. Out of curiosity, he reworked the equations to see how they‘d look in five, rather than four, dimensions. The exercise yielded an extra set of equations, which were the same as James Clerk Maxwell’s equations for the electromagnetic field. His idea of a fifth dimension had produced a mathematical unification of gravity and electromagnetism.

Since Kaluza’s calculations yielded an extra set of equations with Einstein’s formulas for gravity, it’s logical to assume that extending the number of known dimensions to five might also extend E=mc^2. This conclusion is supported by the paper “Did Einstein prove E=mc^2” which states, “The mass–energy formula cannot be derived by Einstein's 1905 argument, except as an approximate relation …” (Ohanian, H.C. Studies in History and Philosophy of Modern Physics Part B: Volume 40, Issue 2, May 2009, Pages 167-173) A more precise relation might be obtained in the following way –

The wave-particle duality of quantum mechanics can be described by starting with v = fλ (wave velocity equals frequency times Greek letter lambda which denotes wavelength). Velocity (speed in a constant direction) of a collection of particles like a car equals distance divided by duration. Since distance is a measure that has to do with space while duration is a measure that has to do with time, it equals space divided by time. (Greene, B. in "Speed", part of his "Space, Time and Einstein" course at http://www.worldscienceu.com/courses/1/elements/YhF9pw) Gravitational and electromagnetic wave motion (space-time motion) travels at c, the speed of light ie

(v= fλ) = distance/duration = space/time = c

A particle's velocity, whether the particle be a boson or fermion, is directly dependent on its energy – so it may be said that

E = (v=fλ) = distance/duration = space/time = c

This is not quite right since c represents energy alone, and space-time deals with mass-energy, so it's better to say

E = (v=fλ) = distance/duration = space/time = mc

What about the 2 in E=mc^2? In later papers Einstein repetitively stressed that his mass-energy equation is strictly limited to observers co-moving with the object under study, and co-movement may be represented by the exponent 2.

If Kaluza and the 5th dimension can extend E=mc^2, can they also extend photon-graviton interaction to the two nuclear forces. This would be done by the quantum spin of the photon (one) and the spin of the graviton (two) producing, via Hawking, * 1 +1/2 -1/2=the spin of the weak and strong forces’ bosons (one).

* Professor Stephen Hawking writes, (Hawking, S.W. A Brief History of Time. Bantam Press, pp.66-67) -

"What the spin of a particle really tells us is what the particle looks like from different directions."

Spin 1 is like an arrow-tip pointing, say, up. A photon has to be turned round a full revolution of 360 degrees to look the same.

Spin 2 is like an arrow with 2 tips - 1 pointing up, 1 down. A graviton has to be turned half a revolution (180 degrees) to look the same.

Spin 0 is like a ball of arrows having no spaces. A Higgs boson looks like a dot: the same from every direction.

Spin ½ is logically like a Mobius strip, though Hawking doesn’t specifically say so. This is because a particle of matter has to be turned through two complete revolutions to look the same, and you must travel around a Mobius strip twice to reach the starting point.

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(Mobius Matrix and Our Familiar Dimensions) Width a is perpendicular to the length (b or e) which is perpendicular to height c. How can a line be drawn perpendicular to c without retracing b’s path? By positioning it at d, which is then parallel to (or, it could be said, at 180 degrees to) a. d is already at 90 degrees to length b and height c. d has to be at right angles to length, width and height simultaneously if it's going to include the Complex Plane's vertical "imaginary" axis in space-time (the "imaginary" realm is at a right angle to the 4 known dimensions of space-time, which all reside on the horizontal real plane). In other words, d has to also be perpendicular to (not parallel to) a. This is accomplished by a twist, like on the right side of the Mobius strip, existing in the particles of matter composing side a. In other words, a fundamental composition of matter is mathematics' topological Mobius, which can be depicted in space by binary digits creating a computer image. The twist needs to be exaggerated, with the upper right of the Mobius descending parallel to side "a" then turning perpendicular to it at approximately the level of the = sign, then resuming being parallel. Thus, 90+90 (the degrees between b & c added to the degrees between c & d) can equal 180, making a & d parallel. But 90+90 can also equal 90, making a & d perpendicular. (Saying 90+90=90 sounds ridiculous, but it has similarities to the Matrix [of mathematics, not the action-science fiction movie] in which X multiplied by Y does not always equal Y times X. The first 90 plus the second 90 does not always equal the second 90 plus the first 90 because 90+90 can equal either 180 or 90).

90+90=90 is transferred from the quantum world to the macroscopic by 1+1=1 because this second equation is useful in the practical, everyday experience of humans and animals. Referring back to the spin of the weak and strong forces’ bosons, the Mobius Matrix’s 90+90=90 becomes not 1+1=1 but 1 +1/2 -1/2 = 1. The compactification of Kaluza’s 5th dimension by Oskar Klein, as well as the compactification of string theory in which four dimensions of spacetime are macroscopic and the others form a compact manifold too small to be observed, may be represented by spin 2 becoming (+1/2 -1/2).

The origin of structure in the universe is one of the greatest cosmological mysteries even today. Extended topological objects such as monopoles, strings and domain walls may play a fundamental role in the formation of our universe. Phase transitions in the early universe can give rise to these topological defects. A topological defect is a discontinuity in the vacuum and can be classified according to the topology of the vacuum manifold. Monopoles are point like topological defects and are formed where M contains surfaces which cannot be continuously shrunk to a pointy i.e. when π2 (M) ≠ I. ( M is the vacuum manifold ) [1] one of most important works about Abelian gauge theories was due to the P. M. Dirac many years ago, who proposed a new solution to the Maxwell equations. His new solution for the vector potential corresponds to a point-like magnetic monopole with a singularity string running from the particle’s position to infinity [2]

[1] F. Rahaman, S.Mal and P. Ghosh; A study of global monopole in Lyra geometry

[2] A. L. Cavalcanti de Oliveira ∗ and E. R. Bezerra de Mello; Kaluza-Klein Magnetic Monopole in Five-Dimensional Global Monopole Spacetime

Some time ago I saw a program where Lee Smolin talked about how Einstein was originally interested, at least some what, with Kaluza-Klein theory. But, after some time he became disinterested in Kaluza-Klein and other extra spatial dimension theories. Smolin claimed Einstein thought about the possibility of extra spatial dimensions, but rejected it on some fundamental principles/insights. I've tried, so far unsuccessfully, to find Einstein's article/text to which Smolin referrers. Does anyone know Einstein's reasoning/insight against extra spatial dimensions? Or, perhaps, the article that Smolin refers to?

There is a significant challenge in higher-dimensional theories regarding how to render the extra dimensions unobservable. A commonly employed approach involves assuming that the extra dimensions are compact and small. However, we can sidestep the necessity for compactification by postulating that spacetime is a subspace of a multidimensional configuration space—specifically, the space of possible matter configurations in 4D spacetime. Instead of expressing physics in spacetime, we can articulate physics within the configuration space.

A potential avenue in this direction was explored in my talk titled "Extending Physics to Clifford Space: Towards the Unification of Particles and Forces, Including Gravity." I delivered this talk as part of the lecture series "Octonions, Standard Model, and Unification," held from February 24 to December 15, 2023. You can find more details about the series here: https://hyperspace.uni-frankfurt.de/2023/02/10/octonions-standard-model-and-unification-online/

The video recordings of these lectures can be accessed at https://www.youtube.com/playlist?list=PLu4STGsfbix-_0BMOtpiH-_hOnBb2Xh5C. Specifically, the video recording of my lecture is available at https://www.youtube.com/watch?v=lsYKz_uMb0c.

In a trial to extend the Bohm interpretation of the quantum mechanics to photons,

D. Bohm, B. Hiley, and P. N. Kaloyerou, "A causal interpretation of quantum fields", Phys.Rep. 144, no. 6, page 349, (1987),

the authors wrote the density of Lagrangian of a scalar field ϕ(x),

(1) L = ½(∂ϕ/∂t)² - ½(∇ϕ)²

then explained that they take ∂ϕ/∂t as canonical momentum

(2) π(x) = ∂ϕ/∂t.

It is a choice that seems to me strange, for a couple of reasons:

- the Lagrangian is defined as T - V, where T is the kinetic energy and V the potential energy.

- however, ∂ϕ/∂t is connnected in the Klein-Gordon equation with another operator, the energy operator i∂ϕ/∂t, which contains also the factor i, otherwise the result is complex, not real.

- if the kinetic energy is defined as ½(∂ϕ/∂t)², then, what can be the other term, ½(∆ϕ)² ? According to the definition of the Lagrangian, it should represent potential energy. However, ∇ is connected with another operator, the linear momentum, -i∇.

Does somebody understand the choice (2)? Did somebody encounter a similar choice?

Can someone please help how to write Klein-Gordan equation for massless particles and also its action or Euler–Lagrange equation for massless particles?

I need to derive the Klein-Gordon equations of the complex Higgs doublet (see the picture). Here ϕ1,2 and θ1,2 are real functions of time. I'm not sure if there are four, i.e. one for each function or only two (for ϕ₁ and ϕ₂).

The topic considered here is the Klein-Gordon equation governing some scalar field amplitude, with the field amplitude defined by the property of being a solution of this equation. The original Klein-Gordan equation does not contain any gauge potentials, but a modified version of the equation (also called the Klein-Gordon equation in some books for reasons that I do not understand) does contain a gauge potential. This gauge potential is often represented in the literature by the symbol A_i (a four-component vector). Textbooks show that if a suitable transformation is applied to the field amplitude to produce a transformed field amplitude, and another suitable transformation is applied to the gauge potential to produce a transformed gauge potential, the Lagrangian is the same function of the transformed quantities as it is of the original quantities. With these transformations collectively called a gauge transformation we say that the Lagrangian is invariant under a gauge transformation. This statement has the appearance of being justification for the use of Noether’s theorem to derive a conservation law. However, it seems to me that this appearance is an illusion. If the field amplitude and gauge potential are both transformed, then they are both treated the same way as each other in Noether’s theorem. In particular, the theorem requires both to be solutions of their respective Lagrange equations. The Lagrange equation for the field amplitude is the Klein-Gordon equation (the version that includes the gauge potential). The textbook that I am studying does not discuss this but I worked out the Lagrange equations for the gauge potential and determined that the solution is not in general zero (zero is needed to make the Klein-Gordon equation with gauge potential reduce to the original equation). The field amplitude is required in textbooks to be a solution to its Lagrange equation (the Klein-Gordon equation). However, the textbook that I am studying has not explained to me that the gauge potential is required to be a solution of its Lagrange equations. If this requirement is not imposed, I don’t see how any conclusions can be reached via Noether’s theorem. Is there a way to justify the use of Noether’s theorem without requiring the gauge potential to satisfy its Lagrange equation? Or, is the gauge potential required to satisfy that equation without my textbook telling me about that?

KG eq. (p²+(m+S)² )=(E-V)²