Zero stands for emptiness, for nothing, and yet it is considered to be one of the greatest achievements of humankind. It took a long stretch of human history for it to be recognized and appreciated [1][4]. In the history of mathematics considerable confusion exists as to the origin of zero. There can be no unique answer to the query, "Who first discovered the zero?", for this may refer to any one of several related but distinct historical issues† [2]. A very explicit use of the concept of zero was made by Aristotle, who, speaking of motion in a vacuum, said "there is no ratio in which the void is exceeded by body, as there is no ratio of zero to a number” [3][2]*. He apparently recognized “the Special Status of Zero among the Natural Numbers.”
If we believe that zero is explicitly expressed mathematically, whether in number theory, algebra, or set theory, is the meaning of zero also clear and unified in the different branches of physics? Or can it have multiple meanings? Such as:
1）Annihilation——When positive and negative particles meet [5][6], e+e-=γ+γ'，the two charges disappear, the two masses disappear, and only the energy does not disappear or increase; the momentum of the two electrons, which was 0, now becomes the positive and negative momentum of the two photons. How many kinds of zeros exist here, and what does each mean?
2）Double-slit interference—— The interference pattern in Young's double slit experiment, what exactly is expressed at the dark fringe? And how should it actually be understood? For light waves, it can be understood as the field canceling due to destructive interference and presenting itself as zero. For single photons, single electrons [7], physics considers it to be a probabilistic statistical property [12]. This means that in practice, at the dark fringes of theoretical calculations, the field will also be likely not to be zero‡.
3）Destructive interference——In Mach–Zehnder interferometer [8]，there's always been a question of where the energy in the destructive interference arm went [9]? There seems to be an energy cancellation occurring.
4）Anti-reflection coatings——By coating [10], the reflected waves are completely canceled out to achieve the purpose of increasing transmission.
5）Nodes of Standing Waves——In optical resonant cavity, Laser Resonator. " The resonator cavity's path length determines the longitudinal resonator modes, or electric field distributions which cause a standing wave in the cavity "[13]. The amplitude of the electromagnetic field at the node of the standing wave is zero, but we cannot say that the energy and momentum at this point are zero, which would violate the uncertainty principle.
6）Laser Beam Mode——The simplest type of laser resonator modes are Hermite-Gaussian modes, also known as transverse electromagnetic modes (TEMnm), in which the electric field profile can be approximated by the product of a Gaussian function with a Hermite polynomial. TEMnm，where n is the number of nodes in x direction, m is the number of nodes in y direction [14].
7）Nodes of the Wave Function——Nodes and ends of the Wave Function Ψ in a square potential well have zero probability in quantum mechanics‡ [11]。
8）Pauli exclusion principle—— Fermions are antisymmetric，Ψ(q1,q2)=-Ψ(q1,q2), so Ψ(q1,q2)=0；Here a wave function of zero means that "field" is not allowed to exist, or according to the Copenhagen interpretation, the wave function has zero probability of appearing here?
9）Photon——zero mass, zero charge.
10）Absolute vacuum——Can it be defined as zero energy space?
11）Absolute temperature 0K——Is the entire physical world defined as a zero energy state except for photons?
12）Perfect superconductor—— "The three 'big zeros' of superconductivity (zero resistance, zero induction and zero entropy) have equal weight and grow from a single root: quantization of the angular momentum of paired electrons" [15].
13）......
Doesn't it violate mathematical principles if we may interpret the meaning of zeros in physics according to our needs? If we regard all zeros as energy not existing, or not allowed to exist here, does it mean that energy must have the same expression? Otherwise, we cannot find a unified explanation.
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Notes
* Ratio was a symmetrical expression particularly favored by the ancient Greeks.
† Symbols(0,...), words (zero, null, void, empty, none, ...), etc..
‡ Note in particular that probability itself is defined as a probability, not an exact value. For example, a probability of 0.5 can occur in physical reality as 0.49999999999, and it is almost never possible to have an accurate probability value such as 0.5. This means that there is no probability value that never occurs, even if the probability is theoretically 0. It is against the principle of probability to assume that a probability of zero means that it will never occur in reality.
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References
[1] Nieder, A. (2016). "Representing something out of nothing: The dawning of zero." Trends in Cognitive Sciences 20(11): 830-842.
[2] Boyer, C. B. (1944). "Zero: The symbol, the concept, the number." National Mathematics Magazine 18(8): 323-330.
[3] the Physics of Aristotle;
[4] Boyer, C. B. (1944). "Zero: The symbol, the concept, the number." National Mathematics Magazine 18(8): 323-330.
[5] https://www.researchgate.net/post/NO8Are_annihilation_and_pair_production_mutually_inverse_processes
[6] https://www.researchgate.net/post/NO22Symmetry_Invariance_and_Conservation_3-Are_Annihilation_and_Pair_Production_a_Supersymmetric_relation
[7] Davisson, C. and L. H. Germer (1927). "Diffraction of Electrons by a Crystal of Nickel." Physical Review 30(6): 705-740.
[8] Mach, L., L. Zehnder and C. Clark (2017). The Interferometers of Zehnder and Mach.
[9] Zetie, K., S. Adams and R. Tocknell (2000). "How does a Mach-Zehnder interferometer work?" Physics Education 35(1): 46.
[10] https://en.wikipedia.org/wiki/Anti-reflective_coating
[11] Chen, J. (2023). From Particle-in-a-Box Thought Experiment to a Complete Quantum Theory? -Version 22.
[12] Born, M. (1955). "Statistical Interpretation of Quantum Mechanics." Science 122(3172): 675-679.
[13]
[14] "Gaussian Beam Optics." from https://experimentationlab.berkeley.edu/sites/default/files/MOT/Gaussian-Beam-Optics.pdf.
[15] Kozhevnikov, V. (2021). "Meissner Effect: History of Development and Novel Aspects." Journal of Superconductivity and Novel Magnetism 34(8): 1979-2009.
Dear Fan,
Perhaps your question is too long to be properly addressed, but I will try my best. I understand that you may have problems with the way in which the concept of zero is applied in physics, rather than with the mathematical concept of zero.
If I have understood correctly, you may have trouble understanding the concept of quantum superposition. A quantum system interacts in ways that can be explained with superposition of different discrete states. Measurements of quantum systems give a statistical result corresponding to any one of the possible states appearing at random. The fact that quantum physics is probabilistic is not easy to grasp, but is key to answer some of your questions.
Any experiment only involving photon detection/observation (e.g. two-slit, Mach-Zehnder, and so on) is basically quantum, because free photons are pure quanta of light, unlike electrons in an iron bar, whose behavior may or may not be microscopic, depending on how you look at them.
That said, let me add that that photons are bosons (mediators) in quantum interactions, and unlike other bosons, their mass is null. If you are implicitly asking why the photon mass is null, it is because otherwise they could not move at light speed, which we know they do. You can safely say that any particle moving at light speed has zero mass.
As to 10) and 11):
10）Absolute vacuum——Can it be defined as zero energy space? --You can set the zero energy state/point wherever you want, and then start counting. Vacuum energy can be experimentally observed in various phenomena such as spontaneous emission, the Casimir effect and the Lamb shift. Using the upper limit of the cosmological constant, the vacuum energy of free space has been estimated to be 10−9 joules (10−2 ergs), or ~5 GeV per cubic meter. However, in quantum electrodynamics, consistency with the principle of Lorentz covariance and with the magnitude of the Planck constant suggests a much larger value of 10,113 joules per cubic meter. This huge discrepancy is known as the cosmological constant problem or, colloquially, the "vacuum catastrophe".
11）Absolute temperature 0K——Is the entire physical world defined as a zero energy state except for photons? --If you mean the CMB, it has a thermal black body spectrum at a temperature of 2.72548±0.00057K. The absolute zero (no entropy) is set at -273.15C. The CMB has been there everywhere and everywhen. And it may have played a bigger role in all physical processes (to explain the vacuum catastrophe, the Landau pole, the cosmological redshift, and so on) than expected. But that is another story...
Dear Jan A. Eidissen Posada
Thank you very much for your patient, professional interpretations!
My intention was that the interpretation of the meaning of 0 in physics should not go beyond math. The 0's that appear in physics, if they can appear, no matter what dimension follows them, energy, field strength, volume, time ...... , they should ultimately point to the same meaning. It may be important to try to relate all the zeros more deeply, and to exclude zeros of unknown significance.
Best Regards, Chian Fan
Dear Chian Fan,
To add some comments, in case they can be helpful, the quantum vacuum should not to be identified with an empty set in math. It is the state with the lowest possible energy density, but not a simple empty space.
In fact, according to several experiments, the vacuum energy of free space is not zero (with the Casimir effect being most elucidatory), as I mentioned. The question would be if empty space is physically feasible in our universe. And the CMB points right to the opposite. Even if no massive particles are present at some regions of the universe -which is far from clear-, the energy of "empty space" (in the interstellar medium) would not be zero, because the CMB permeates space and time. And creating interstellar conditions with no massive particles or CMB in a lab is impossible (the Casimir effect was not measured in a perfect vacuum).
Also, scales in physics are not unimportant. The absolute zero temperature is 0 in the Kelvin scale, but is also -273.15 degrees Celsius, which is a negative number, and means exactly the same temperature (at which there would be no heat flow or entropy).
So, I would not become too obsessed with equating all zeroes in physics and math.
Best regards,
Jan
Dear Jan A. Eidissen Posada
I apologize that my memory may be faulty for not responding to you in a timely manner.
I understand that you represent the views of mainstream physics. But I still think it's potentially regrettable to easily write off the significance of their consistency.
It may be true that there is no vacuum in physics, but that does not mean that there must be a vacuum energy bound up with space or time. Physics' theoretical explanations and experimental verification of vacuum energy are incomplete and therefore hardly convincing, and while it can keep physics going for a while, it may be a question of how far and whether it will be an obstacle. I believe that all energies are the same, whether it is "vacuum energy" or "dark energy", as long as they are energies there must be a normal transfer and exchange, even if they are the dominant ones, and electromagnetic radiation and matter are the derivative things.
Anyway, thank you for your expert opinion.
Best Regards， Chian Fan
Regarding "I understand that you represent the views of mainstream physics.", it is much more likely than not that you are wrong, but time will say.
Whether my viewpoints are coincident with yours, that is another question.
As to: "Physics' theoretical explanations and experimental verification of vacuum energy are incomplete and therefore hardly convincing, and while it can keep physics going for a while, it may be a question of how far and whether it will be an obstacle. I believe that all energies are the same, whether it is "vacuum energy" or "dark energy", as long as they are energies". This sounds vaguely familiar to me. If you can convince me that you have solid evidence supporting these statements, surely we can talk.
Best regards,
Jan
PS: My take is that vacuum energy obviously exists (and is small), but the so-called "dark energy" does not exist. It just reflects our misunderstanding of particle physics.
Dear Chian Fan
Although I agree with Jan A. Eidissen Posada , I have some remarks about the meaning of zero in physics.
In phenomenological physics zero is an easy concept. It means the non-existence of a phenomenon (and its multiple) in relation to the existence of the phenomenon within a certain region or even a distinct position in the universe.
The law of conservation of energy and the equivalence of matter and free energy (E = m c2) eroded this simple concept. Because the disappearing of a phenomenon didn’t mean that the amount of energy of the phenomenon was “deleted” to zero. It was only transformed into another energy configuration with different geometrical properties (e.g. an expanded volume like E = m c2 predict).
The consequence is that zero seems to be the point of balance within a symmetry of dynamical changes. Although all the properties of the symmetry cannot be 100% symmetrical because that results in a 100% static universe.
In modern quantum field theory the general concept is threefold. First, all the basic quantum fields together tessellate the volume of the universe. Second, every phenomenon is “created” by the basic (elementary) properties of these quantum fields. Thirdly, the basic quantum fields together represent a rest frame in relation to all the observable and detectable phenomena in the universe.
Of course these “creations” are not from nothing (zero). The “creations” are transformations under one or more invariances within the structure of the basic quantum fields (the rest frame). In line with the accepted experience that all energy phenomena represent dynamical mutual relations (no absolute properties).
The draw back of physics as a science is its culture (sometimes it is like a pseudo-religion). The absence of nothing (zero) in our universe was already described by ancient Greek philosophers (the Eleatic school of philosophy). But physics was/is mostly focused on the observable/detectable phenomena. We use a doctrine – the scientific method – that describes the process of the reverse engineering of physical reality. So actually, it is the physics method, not the scientific method. But we pretend that physics is at the top of scientific knowledge. So every theorist has to confront libraries full of papers with limited scientific meaning.
With kind regards, Sydney
Dear Jan A. Eidissen Posada and Sydney Ernest Grimm
Your points are quite profound, touching on the nature of physics and worthy of deeper thinking.
At the most basic level, I have always believed that there exists something that expresses everything. The most appropriate choice for such a thing is energy, which is universal and exchangeable. What we see as the material, radiation universe is likely to be a tiny fraction of it. However, I have not yet to figure out what conditions prevent them from interacting with each other in a vacuum, thus calming them down.
Best Regards, Chian Fan
Dear Chian Fan,
It's very kind of yours, thanks. And excuse me if I got a bit over the top in my last answer.
Since the 1980s, being on the mainstream side of HEP almost unavoidably meant working in string theory. I have always tried to take theoretical and experimental evidence on physical phenomena for what it's worth, and for this reason I have never worked in ST. And I staunchly support the belief that the Standard Model of physics is almost complete, lacking some finishing touches, which can be achieved within its realm -- even if this sounds opinionated.
Given that stringy ideas have dominated academic research in the Western hemisphere since the 1980s, I'm quite on the non-mainstream side in this regard. And my ideas as to how the SM could be completed might be considered non-mainstream on their own too.
That said, I would recommend you to think a little bit 'a la Zwicky' if you are wondering if dark energy exists (the Casimir effect is sound evidence that vacuum energy exists and is small). Zwicky's tired light mechanism was obviously wrong, but you can ask yourself if some other mechanism could redshift the signals that we receive from standard candles, supernovae, and so on, without suffering the problems of Zwicky's 'naive' proposal.
Perhaps some quantum mechanism could provide an (altenative to dark energy) explanation for large part of that cosmological redshift. Especially if it's the same mechanism that allows you to slightly improve theoretical calculations of the anomalous magnetic moment of the electron, without renormalization issues, or replicate all cross sections in QED. One specific physical idea may lay behind it, and there are some mathematical rules/tricks to follow, but apart from that, nothing too sophisticated, although this physical idea might not be as self-evident as it should. Clearly not one an AI will suggest...
Regards,
Jan Eidissen
The clue of deeper meaning of "zero" was described by @Fred-Rick Schermer in [https://www.researchgate.net/post/No28The_Relation_Between_Mathematics_and_Physics_4-Could_All_Physical_Constants_be_Unified_to_a_set_of_Fundamental_Constants_of_Nature]. I moved it here for the sake of completeness of this thread. It is a key concept in his book The Proof of Nothing [https://www.researchgate.net/publication/377747282_The_Proof_of_Nothing].
One can accept zero to be part of every system, or of every outcome.
This has led to Intuitive Mathematics in which there is always a third way. For instance, next to yes and no, there is 'maybe' as well.
So while there are in total just two answers (for instance, male and female), there is still a third option. Take a rock, for instance, and one has to return the question male or female with two zeros. The rock is real, the male/female system 'just' a system, and it must therefore contain the zero option, the way out. The system does not apply to the rock, there are third outcomes.
No system can return a completeness at the overall level or reality.
Zero is therefore not the unifying principle either. Zero declares that there are no unified principles (at the largest level).
It is with zero that we can have the space to consider the decimal system and the binary system. Both systems can do anything, mathematically, the other system can do as well. And yet they are not the same systems. All thanks to zero that did allowed other options to come into being just as easily as any other system.
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