Complex numbers are involved almost everywhere in modern physics, but the understanding of imaginary numbers has been controversial.
In fact there is a process of acceptance of imaginary numbers in physics. For example.
1) Weyl in establishing the Gauge field theory
After the development of quantum mechanics in 1925–26, Vladimir Fock and Fritz London independently pointed out that it was necessary to replace γ by −iħ 。“Evidently, Weyl accepted the idea that γ should be imaginary, and in 1929 he published an important paper in which he explicitly defined the concept of gauge transformation in QED and showed that under such a transformation, Maxwell’s theory in quantum mechanics is invariant.”【Yang, C. N. (2014). "The conceptual origins of Maxwell’s equations and gauge theory." Physics today 67(11): 45.】
【Wu, T. T. and C. N. Yang (1975). "Concept of nonintegrable phase factors and global formulation of gauge fields." Physical Review D 12(12): 3845.】
2) Schrödinger when he established the quantum wave equation
In fact, Schrödinger rejected the concept of imaginary numbers earlier.
【Yang, C. N. (1987). Square root of minus one, complex phases and Erwin Schrödinger.】
【Kwong, C. P. (2009). "The mystery of square root of minus one in quantum mechanics, and its demystification." arXiv preprint arXiv:0912.3996.】
【Karam, R. (2020). "Schrödinger's original struggles with a complex wave function." American Journal of Physics 88(6): 433-438.】
The imaginary number here is also related to the introduction of the energy and momentum operators in quantum mechanics：
https://www.researchgate.net/post/What_are_the_roots_of_the_energy_operator_ih_t_and_the_momentum_operator_ih_x
Recently @Ed Gerck published an article dedicated to complex numbers:
Preprint There Are No Complex Numbers: a 500-year prelest
Our question is, is there a consistent understanding of the concept of imaginary numbers (complex numbers) in current physics? Do we need to discuss imaginary numbers and complex numbers ( dual numbers) in two separate concepts.
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2023-06-19 补充
On the question of complex numbers in physics, add some relevant literatures collected in recent days.
1) Jordan, T. F. (1975). "Why− i∇ is the momentum." American Journal of Physics 43(12): 1089-1093.
2）Chen, R. L. (1989). "Derivation of the real form of Schrödinger's equation for a nonconservative system and the unique relation between Re (ψ) and Im (ψ)." Journal of mathematical physics 30(1): 83-86.
3) Baylis, W. E., J. Huschilt and J. Wei (1992). "Why i?" American Journal of Physics 60(9): 788-797.
4）Baylis, W. and J. Keselica (2012). "The complex algebra of physical space: a framework for relativity." Advances in Applied Clifford Algebras 22(3): 537-561.
5）Faulkner, S. (2015). "A short note on why the imaginary unit is inherent in physics"; Researchgate
6）Faulkner, S. (2016). "How the imaginary unit is inherent in quantum indeterminacy"; Researchgate
7）Tanguay, P. (2018). "Quantum wave function realism, time, and the imaginary unit i"; Researchgate
8）Huang, C. H., Y.; Song, J. (2020). "General Quantum Theory No Axiom Presumption: I ----Quantum Mechanics and Solutions to Crisises of Origins of Both Wave-Particle Duality and the First Quantization." Preprints.org.
9）Karam, R. (2020). "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level." American Journal of Physics 88(1): 39-45.
It is interesting that imaginary numbers appear as a loss of probability in quantum mechanics. It appears in the calculation of relativistic mass as a particle that may travel faster than the speed of light. Imaginary numbers in physics mean something and it can not simply be tossed away.
Dear Preston Guynn
Thank you very much for your enlightening answer, but my understanding of your example is that the concept of integration is not the same as "area"; "mathematical area" is not the same as "physical area". We need to be aware of the figures that appear in my question: Weyl is one of the greatest mathematicians of the 20th century, and Yang and Schrödinger are the greatest physicists. Therefore, the understanding of imaginary numbers in physics cannot be answered simply by √-1.
I hope I'm the one who didn't fully grasp your point.
Best Regards, FAN
Dear Chian Fan
As we all know, mathematics is a "language".
I observe that two types of persons take interest in physics, pure mathematicians and experimental physicists.
What differentiates both types is that validity of reasoning is provided in the case of experimental physicists by "numerical resolution" of whatever equation can be drawn from physically collected data that experimental physicists use in describing what they observe, and validity of logical derivations from sets of axiomatic postulates in the case of pure mathematicians.
One of the major difficulties in fundamental physics is the very power of mathematics as a descriptive language. If care is not taken to avoid as much as possible axiomatic postulates, an indefinite number of theories can be elaborated with full mathematical support that can always become entirely self-consistent with respect to the set of premises from which each theory is grounded. But the very self-consistency of all well thought out theories is so appealing to our rational minds that it renders very difficult the requestioning of the grounding foundations of such beautiful and intellectually satisfying structures and consequently the identification of possibly inappropriate axiomatic assumptions.
Experimental physicists adapt the available math as well as they can in their attempts at mathematically describing what they observe from the data they collected – of which i never is an element – while pure mathematicians explain what logically comes out of whatever sets of axiomatic premises that they chose to underlie their worldview.
From what I understand, √-1 just happened to be part of the mathematical toolset that Schrödinger had at his disposal in trying to mathematize how to account for the stationary resonance state that de Broglie had discovered that the electron is captive of when stabilized in the hydrogen atom ground state, a resonance frequency to which those of all other metastable orbitals of the hydrogen atom and resulting emitted bremsstrahlung photons are related by the well established sequence of integers that de Broglie provided in his 1924 thesis.
Best Regards, André
For any mathematical concept one must see if there is a direct or indirect relationship. One can get some idea of reality when one uses a fraction to say that there are 2.5 children per family in the US. However, there is no such thing as .5 children. But children do exist. One must go back to the direct relevance of counting and restate the fraction as there are 250,000 children per the100,000 families in the US, and there is no problem. One can see that children exist (2) (PDF) Quantum wave function realism, time, and the imaginary unit i. Available from: https://www.researchgate.net/publication/322386956_Quantum_wave_function_realism_time_and_the_imaginary_unit_i [accessed May 29 2023].
this paper above explains the direct relevance of the imaginary unit as a vector direction for time.
Dear André Michaud
Thank you very much for your philosophical explanation and advice.
I am reading your paper "Field Equations for Localized Individual Photons and Relativistic Field Equations for Localized Moving Massive Particles " and Paul Marmet's paper "Fundamental Nature of Relativistic Mass and Magnetic Fields".[http://www.newtonphysics.on.ca/magnetic/index.html]
I appreciate the insightful and valuable ideas and look forward to getting help.
Best Reragds， FAN
Dear Preston Guynn
Figures means VIP, sorry this may be an inappropriate way to discuss the matter.
Thank you very much for your patience, I think I understand what you mean by "mapping", QPSK modulation in communication systems is probably a typical "physical-mathematical" mapping relationship. It is mathematically expressed as exp(iωt) for the physical quadrature signal. But my concern is what exactly is mapped to i in quantum mechanics, quantum field theory, or what are a and b when a phenomenon can be described as s=a+ib? And what is the meaning of i at this point?
Best Reragds， FAN
Chian Fan
Dear Fan,
Glad that you are taking interest in electromagnetic mechanics.
The papers that you mention that you are reading were key in triggering the sequence of analyses that led to the resumption and conclusion of Wilhelm Wien's 1901 project to establish electromagnetism as the common foundation to both kinematic and electromagnetic mechanics for the establishment of differential equations that would be applicable to each from a common basis, instead of kinematic mechanics that was chosen in 1907.
Within a few days will be published the final paper in the project, titled "Introduction to Synchronized Kinematic and Electromagnetic mechanics", that also includes the related description of the four stationary intensity levels of the trispatial vector field and of their vector complexes.
When available within a few days, a direct link will be provided to it as the first entry in this Index:
Article INDEX -Electromagnetic Mechanics (The 3-Spaces Model)
If you have any questions on any aspect of this project, I will remain available to provide as appropriate answers as I can manage.
Best Regards, André
That the concept of the imaginary number (count of standard measurements such as meters) works in current models is indicating that there is some other (unknown or unaccounted) physical presents that should be included for a physical understanding. That is, a Kuhn type paradigm shift is needed in the model.
So, the quantum wavefunction is not observable (real, measureable) but the square of it is measureable implies there is a parameter (that was termed imaginary) that has an influence on the measurement. Hence, the ``measurement problem''.
For example, the Young's experiment was thought to suggest light is a wave. Later, the photoelectric demonstrated light is a particle (photon). Waves are described using the imaginary factor. Then de Broglie and Bohm suggest some kind of guiding wave (still with imaginary unit). But there is no source for the low intensity (one photon in the experiment) interference experiments. Even with more difficulty does the imaginary unit needed for the Hodge experiment (1). Note the Fraunhofer approach introduces 5 assumptions with a sin() function (no i) including the plane from which measurement are taken. His new assumptions for the single slit may be viewed as replacing the i in other (Sommerfield's for instance) analysis. But Frauhnhofer fails in the double slit and the Hodge experiment. So, the STOE changes the ``source'' to be the photon causing waves in a continuous substance (plenum like Descartes') descibed with a cos() function generator of the photon. This total elimination of i yields measurements/observations explaining Young's and Hodge's experiment. NO i-based model does this.
Interference Experiment with a Transparent Mask Rejects Wave Models of Light
Optics and photonics journal vol. 9, No. 6 jun 2019
https://www.scirp.org/journal/paperinformation.aspx?paperid=93056
https://doi.org/10.4236/opj.2019.96008
https://www.youtube.com/watch?v=qFDB-K_sSjU
https://www.youtube.com/watch?v=A07bogzzMEI
Dear Martin Hay
Correct, the multiplication with i is usually connected with a phase shift of 90°. Once I assembled a circuitry to generate two rectangular impulses, which are shifted about 90° against each other. Besides a binary counter I needed an XOR gate.
what a coincidence...
To me, the imaginary number is to realize the Gauge theory as you have mentioned. It arbitrarily set a zero energy reference point for a system and then distributes some of the energy which is real into the imaginary dimension often as momentum. This is a convenient operation because it is very easy to maintain the modules of the complex number as a constant throughout space and time using the Euler expression avoiding complicated matrix operation. But at the end of the calculation, we must make sense of the physical meaning of the complex number and find out where do we initially set the reference point. For example, a system with a total energy of 1 eV versus the absolute zero energy level but we set the energy level zero at 0.5 eV. Thus the wavefunction will have 0.5 eV in the imaginary part and 0.5 eV in the real part such that the complex number of its momentum will be sqrt(0.5)+-sqrt(-0.5). After calculation and interaction with other waves, we will need to find out if the reference point has changed or maintained which I think is a complicated process and the negative real energy and negative imaginary energy make things even more complicated, i.e., what is the meaning of 0.3+0.6i after a calculation?
For an electromagnetic wave, I agree with many including Dr. André Michaud who has excellent examination into the history of this problem, that it should be dividing half of the energy in the electric field and half of the energy in the magnetic field, i.e. all energy can be interconverted between the two fields just like an electric transformer transfers energy from one side to the other via vibrating electric and magnetic fields. That is, the energy conservation law is maintained and even in the quantum world we cannot break it which the current theories have trouble getting to this point. If you check my publication list there is a preprint proposing an alternative quantum equation to challenge the Copenhagen interpretation in quantum mechanics. Preprint From Particle-in-a-Box Thought Experiment to a Complete Quan...
On the question of complex numbers in physics, add some relevant literatures collected in recent days.
1) Jordan, T. F. (1975). "Why− i∇ is the momentum." American Journal of Physics 43(12): 1089-1093.
2）Chen, R. L. (1989). "Derivation of the real form of Schrödinger's equation for a nonconservative system and the unique relation between Re (ψ) and Im (ψ)." Journal of mathematical physics 30(1): 83-86.
3) Baylis, W. E., J. Huschilt and J. Wei (1992). "Why i?" American Journal of Physics 60(9): 788-797.
4）Baylis, W. and J. Keselica (2012). "The complex algebra of physical space: a framework for relativity." Advances in Applied Clifford Algebras 22(3): 537-561.
5）Faulkner, S. (2015). "A short note on why the imaginary unit is inherent in physics"; Researchgate
6）Faulkner, S. (2016). "How the imaginary unit is inherent in quantum indeterminacy"; Researchgate
7）Tanguay, P. (2018). "Quantum wave function realism, time, and the imaginary unit i"; Researchgate
8）Huang, C. H., Y.; Song, J. (2020). "General Quantum Theory No Axiom Presumption: I ----Quantum Mechanics and Solutions to Crisises of Origins of Both Wave-Particle Duality and the First Quantization." Preprints.org.
9）Karam, R. (2020). "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level." American Journal of Physics 88(1): 39-45.
Perhaps we can ask the authors here to give a concise introduction.
Real and imaginary numbers are just one way of dealing with signals that have amplitude and phase, or amplitude and direction. Such signals need two numbers to describe them. The mathematics is exactly equivalent but it is usually much easier and more concise using complex numbers than trigonometric identities and twice the number of equations.
I guess one my start with some simple example, e.g. damped classical oscillator. The description of it, being solution of second order diff equation ( exponentially decaying some sine + some cosine wave) naturally comes along Euler formula for imaginary exponent.
Another interesting, although a bit of joke thought:
The Schrödinger equation (which is parabolic simiraly to diffusion equation but with imaginary 'diffusion coefficient' ) can be though as diffusion in/from some 'imaginary' wavefunction component into the 'real' one.
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