I believe you are correct that complex numbers are an integral part if physics and not simply a convenience to simplify intermediate calculations where we take the real or imaginary part of the final result as the actual answer. You might note that in constructing the time-dependent Schroedinger wave equation, the time derivative term (i \hbar d\psi / dt) contains the factor i. This was done to allow wave-like solutions while keeping the equation first-order in time so that time-dependent perturbation theory could be developed for calculating transition rates. All other wave equations I have seen are second-order in time. Also, the eigenfunctions of the momentum operator are imaginary exponential functions instead of sin and cos functions. So, in order to complete the modern quantum theory, it was necessary to introduce the concept of complex valued wavefunctions and operators. Subsequently, with new particle discoveries, it became necessary to introduce even more generalized ways of representing states.

Dear Dwight Walsh ,

Thank you for the insight. I agree that in quantum mechanics, complex numbers are necessary, I've tried to illustrate this in a short study

Preprint On complex numbers in quantum mechanics

F. Barzi

Let me answer with a link

https://www.researchgate.net/post/What_is_an_action

F. Barzi ,

the introduction of complex numbers for an useful use in Physics was first performed by OLIVER HEAVISIDE to solve circuits..

Article Oliver Heaviside's electromagnetic theory

Heaviside was an English self-taught electrical engineer, physicist and mathematician, who changed the face of telecommunications, mathematics and science for years to come, up to and well beyond his death. He adapted complex numbers to the study of electrical circuits, coining such terms as inductance, impedance and reluctance, among many others, all in common use today.

Hamilton Quaternions (bi complex) were invented in 1843 and used even to express Maxwell equations in 1870, but with little success due to their "complexity". Heaviside decided to write the Maxwell equations in the vectorial form, the one used nowadays.

F. Barzi ,

The fact that the commutator of the momentum and position operator is proportional to the imaginary unit is easily explained if the operators of the observables correspond to Lie algebras of vector fields. Indeed, the coordinate vector field and the impulse vector field are obtained from each other by rotation, and the imaginary unit corresponds exactly to the Lie algebra of rotations.

Stefano Quattrini On the original formulation of Maxwell in terms of Quaternions and the usual Heaviside formulation, it is not obvious they are equivalent? this question may deserve more attention.

Dear Igor Bayak,

I believe that what you said about Lie algebras is an answer to a How complex numbers enter physics? but the question here is Why they do? the fundamental question is still lacking a satisfactory answer. In other words, Is the world inherently complex? and our inability so far to grasp this nature has frustrated our serious attempts to understand it? Is there an intrinsic complex way to look at the world that does not rely on always going through a rational number to "make sense"?

F. Barzi ,

On the contrary, I tried to answer the question about the reason for the appearance of complex numbers in quantum mechanics, and not about how they appear there. In a nutshell, the reason is that in fact a material point is not a point, but a circle.

F. Barzi ,

>

I did not study at all the Quaternionic version. It would be quite interesting to know where vectorial and Quaternionic differ in their prediction power.

Igor Bayak

You propose that a spinless point particle has some internal structure that needs complex numbers(CN) to explain it. right? (because a particle with spin necessary needs CN). A circle in what space?

F. Barzi ,

Even the symmetries of a particle with a half-integer spin can be described without complex numbers, if we take into account that the algebra of complex matrices has a representation in the form of an algebra of linear vector fields in spaces with a neutral metric. The algebra of two-row matrices is in 4-dimensional space, 4-row matrices are in 8-dimensional space. But to determine the wave function of quantum mechanics, it is not important, but the fact that the seemingly rectilinear motion of a particle actually represents movement on the surface of a 7-sphere, and therefore the particle, with such movement, winds revolutions around a circle.

I suggest that complex numbers must not be used, as with finding that spacetime must be treated 12D, one should only calculate with a^2+b^2=c^2 and not use sin, cos or any other representation of "pi" (i, e, phi, pi ) as a "number".

Preprint No complex numbers in Physics

Dear M.U.E. Pohl ,

Complex numbers in physics is no more a matter of choice or suggestion it is proven that a real quantum mechanics is not equivalent to the standard complex quantum mechanics, like $R^2$ is not equivalent to $C$. They are absolutely necessary. Now we need to build a strong complex intuition about them in a way or another, to go deep into the complex realm, to find what we're missing about the complex nature of reality.

Dear Igor Bayak ,

Could you please be more specific? or point to good references.

F. Barzi ,

complex numbers in circuits and control theory provide the necessary degree of freedom to express internal states of a system which might not be necessarily accessed. For circuits such numbers are direct description of the "stored energy" in inductance and capacitors, a way to account for "potential energy", the two description merge since control theory relies eventually on electronic circuits.

If you require me to refer to suitable literature, then, unfortunately, I can only offer my own writings.

https://www.researchgate.net/publication/329252706_MATHEMATICAL_NOTES_ON_THE_NATURE_OF_THINGS

Igor Bayak ,

You need to reformulate and motivate your ideas more carefully to be intelligible. Using a term such as "metaphysical space" in your writing will throw any serious reader off.

Serious reader. Who is this?

F. Barzi ,

The introduction of complex numbers in physics initially served as a mathematical convenience, but their role has evolved to become increasingly fundamental, indicating that we might indeed be missing a deeper interpretation. In my paper, Article Origin of Heisenberg's Uncertainty Principle

Complex Numbers and Quantum Mechanics: Complex numbers are integral to the formulation of quantum mechanics, particularly in describing the wave functions of quantum states. These wave functions are inherently complex-valued, with both real and imaginary components. The imaginary part, often seen as a mathematical artifact, actually plays a crucial role in capturing the probabilistic nature and the dynamic evolution of quantum systems.

Heisenberg's Uncertainty Principle: In my paper, I argue that the physical origin of Heisenberg's uncertainty principle can be better understood by considering complementary variables, such as position and momentum, as complex numbers. This approach reveals that the lower limit of uncertainty arises from the inherent properties of particles existing in a complex vector space rather than a real vector space.

Particles in Complex Vector Space: A particle's position and momentum are typically measured in a real vector space, but if we consider these variables in a complex vector space, the uncertainties associated with them are not mere artifacts but reflections of the particle's internal vibrations and interactions. The real part of the complex number represents the measurable coordinate, while the imaginary part accounts for the internal dynamics and vibrations of the particle, as shown in the complex plane representation in my paper.

Physical Interpretation of Complex Components: The imaginary components of complex numbers in quantum mechanics can be interpreted as representing the internal, often hidden, aspects of physical reality. For instance, when we measure a particle's position, we capture only the real part of its state, while the imaginary part remains unseen but influences the probabilities of different outcomes. This dual representation provides a more complete picture of the particle's state.

Spacetime and Complex Numbers: Incorporating complex numbers into our understanding of spacetime could offer new insights into the fabric of reality. The curvature and dynamics of spacetime itself might be better represented using complex numbers, where the imaginary components could account for the probabilistic and non-deterministic aspects of quantum phenomena. This perspective aligns with the view that spacetime is not merely a passive stage but an active participant in the quantum dynamics.

Conclusion: The use of complex numbers in physics goes beyond mathematical convenience and suggests a deeper, more fundamental role in describing reality. By considering variables such as position, momentum, energy, and time as complex numbers, we gain a more comprehensive understanding of the limitations and behaviors predicted by Heisenberg's uncertainty principle. This approach could lead to a new interpretation of quantum mechanics and a better grasp of the underlying nature of spacetime.

Dear Bhushan Bhoja Poojary ,

Thank you for your reply. and sharing your important ideas with us. I think this is the way to go: extending by analytic continuation and complexification of known fundamental concept looking for their deeper maybe true significance and I commend you for this. I 'll look at your work for insight, however I want to say that in the process we must not violate the laws of physics for example that quantum mechanics requires observables to be hermitian operators or that not all complex metrics are valid ( https://arxiv.org/pdf/2111.06514 ). Thanks again

Prof. F. Barzi

Yes, it is fundamental to use calculations that output complex values and use them correctly, I must say.

Particularly in problems that involve self-consistent variables (I say this based on my own experience as a researcher).

Best Regards.

In my opinion, Edward Witten is considering very artificial complex spaces (extensions of Einstein's metric space). This method can lead you to the wrong place. On the other hand, there is another method by which the Einstein metric space can be obtained from complex algebra. In this regard, I note that the unit sphere of 8-dimensional space with a neutral metric has the topology R^{4} x S^{3}, and the Lie algebra of vector fields tangent to this sphere generates the algebra of Dirac matrices.

Prof. Pedro L. Contreras E. ,

Thank you for your reply. You mentioned "self-consistent variables" could you give a concrete example?

Dear Igor Bayak ,

I can not understand an algebra generated by another algebra do you mean isomorphic?

Each Clifford algebra has its own Lie algebra, and from a geometric point of view, a Lie algebra generates a corresponding Clifford algebra using vector fields tangent to the hyperspheres of metric spaces.

Igor Bayak,

Thanks a lot for pointing to Clifford algebras.

Dear Prof. F. Barzi, yes I can give an example:

In unconventional superconductors, the elastic scattering cross-section formalism to analyze the phenomenon has a complex solution, but the energy is self-consistent, and the variables change very fast.

It is quite complicated to find a numerical solution, I have worked in the field for 24 years now.

Best Regards.

Dear F. Barzi ,

I corrected the text. Now there is no metaphysical space, which has been replaced by a metaphysical extension of Euclidean space. By the way, there (p. 5 [1]) it is shown that the metaphysical expansion of Euclidean space is equivalent to the complexification of Euclidean space.

(1) (PDF) Mathematical Notes on the Nature of Things (fragment) (researchgate.net)

https://www.researchgate.net/publication/377842836_Mathematical_Notes_on_the_Nature_of_Things_fragment

Dr F. Barzi ,

The introduction of complex numbers in physics initially served as a mathematical convenience, but their role has evolved to become increasingly fundamental. In my paper, "Thesis Emergent Universe from Many Unreal World Interpretation

Complex Numbers and Quantum Mechanics: In this paper, I propose that every particle has a dedicated address in the event horizon, with each address having its own dedicated spacetime fabric. The interaction between true particles and their shadow particles across different spacetime fabrics can be understood using complex numbers. This framework helps explain the wave function collapse and entanglement.

Wave Function Collapse and Complex Numbers: The wave function collapse is a significant quantum phenomenon where a particle's state reduces from a superposition of states to a single state upon measurement. In my interpretation, when a true particle interacts with a shadow particle, they get destroyed and create new particles with new addresses. This process can be described using complex numbers, where the real part represents the measurable state and the imaginary part accounts for the internal dynamics of the particles.

Entanglement and Complex Numbers: Entangled particles share the same address in the event horizon. The use of complex numbers allows us to understand how the measurement of one particle affects the state of its entangled partner instantaneously. The imaginary components of the complex numbers capture the probabilistic nature and the underlying interactions between the entangled particles.

Higher-Dimensional Space: The idea that each particle's address has its own dedicated spacetime fabric suggests a higher-dimensional space where complex numbers play a crucial role. This aligns with the notion that complex numbers are not just mathematical artifacts but fundamental components of physical reality.

Conclusion:

In "Thesis Emergent Universe from Many Unreal World Interpretation

I encourage you to read my paper for a detailed exploration of these concepts and how they contribute to our understanding of quantum mechanics.

In alternative electric current calculations of physics & electrical engineering extensively used !

@Sinan

Only as a convenience or a tool not as a fundamental concept. I am looking at a deeper insight into their connection to the physical world we live in.

I think we were all surprised the first time we got to know quantum mechanics that the squared modulus of the wave function is the probability density of the existence of the particle.

The role of the complex numbers here is strange, but the question here is:

Is there an idea that is deeper and easier to understand, so that entering the squared modulus of the complex number becomes a mathematical result from this idea only to facilitate the calculations?

God willing, I think we can find something deeper and even simple, actually, I put this reason in my paper:

Article A CONCEPT OF UNIVERSAL QUANTUM JUMP

In summary:

The main idea of this paper is that the continuous trajectory of the particle can not exist, so the motion is a sequence of appearances and disappearances events in space and time, so the particle does always jump to move from one position to another.

So when the particle is in position p1 at time t1, where would it be in time t2?

In classical mechanics, the trajectory exists so the least action principle states that:

The path the particle takes between times t1 and t2 is the one for which the action is stationary.

So what is the situation in quantum mechanics?

fortunately, we have a principle that is very close to the classical principle, but in this case, we didn't have any path, we have potential new positions, so in general, the particle has some preferred destinations based on a new quantum action principle named "alike action principle" that ensures the existence of physical harmony within our universe, like for example preventing the particle from easily reaching forbidden locations (guarded by fields of great forces).

Therefore, in general, this new constraint in motion could be valid at multiple positions at the same time, so in general, we have numerous acceptable positions at time t2.

Thus the probability of existence came up in our description of the movement in the quantum world.

We suppose that we have a preferred value of action that we call h (Plank constant), the new action principle called the "alike action principle" states:

"The preferred appearance destination position taken by the particle at time t is the one for which all the remainders due to S/h (for all imaginary paths which lead to this destination) are stationary".

In other words, it is as having the same (or close to each other) remainder after dividing them by h.

For example, if we have two actions (for two paths) to one destination position, the natural function that verifies this principle is:

sin2((π/h)(S1 − S2)).

So after some steps of the calculation, we derive the relation between the probability density of the existence and the squared modulus of the wave function by deriving the path integral formulation of quantum mechanics, for more details please see the paper.

Thanks.

In his highly regarded book visual complex analysis, Tristam Needham makes a compelling argument that the true interpretation on complex numbers is the vector direction of time. pp. 10 -12.

NeedhamVCA.pdf (upjs.sk) . This is one of the best books concerning complex numbers. IMHO.

He also directly relates mobius transformations on the complex plane to Lorentz transformations on pp. 122-123. So does Penrose and Rindler (1984, chapter 1)

I approach the true interpretation of complex numbers as the vector direction of time using a different proof. Article The imaginary unit i as the temporal directional component o...

Best Regards

Greetings. Complex numbers are actually fundamental in nature, fermions/ matter fields, which follows Pauli exclusion principle, cannot be described properly without using them, since the definition of spinors, as irreducible representations of Spin(n) groups, does indeed needs passing from a skew-symmetric property (matrices with real numbers) of Lie algebra generators, describing the group of spacial rotation SO(n), which is applied to vectors only, to the full rotation symmetry group Spin(n), which its Lie algebra, hosts anti-Hermitian generators, which forces us to use pure complex numbers. So we may conclude that skew-symmetric property found in SO(n) is not "complete" in a sense, so it was generalized by the anti-Hermitian property, which allowed for introducing spinors representations on top of vectors.