You just saw a picture of what mass is. Here is how to talk to your teacher about it.
You: “I saw a picture of mass. It said light always goes straight and hides the space behind it. If light gets bent into a loop, it hides the space inside the loop. That trapped light is mass. Is it OK if I think of mass that way?”
Your teacher will probably say something like: “Mass is how much stuff is in something.” That is what the books say. One says: “mass is a measure of the amount of ‘stuff’ (or matter) in something” (OpenStax, College Physics, 4.2).
Your teacher is right. Mass tells you how much stuff there is. But it does not say what the stuff is made of. So you can ask:
You: “What is the stuff made of? My picture says it is light that got trapped in a loop. Is there a different picture I should learn?”
There is one more thing the books say. Mass is also how hard it is to push something. That is called inertia: “The property of a body to remain at rest or to remain in motion with constant velocity is called inertia” (same page). The picture says the same thing in its own way: the loop keeps hiding the same space every time around, and that is why it is hard to push.
You: “On the test, will there be a picture of mass, or just the word?”
Read more: Bridge Narrative (step 3, mass as missing space) on this site. The picture on this page is drawn from it.
You have just read a page where a photon closed into a loop hides a fixed amount of space every cycle, and that hidden amount is the inertial mass. F = ma and E = γmc² both come out of it. Here is how the conversation with your teacher might go.
You: “I’ve been looking at a model of mass: a photon closed into a loop hides a fixed amount of space every cycle, and that hidden total is the inertial mass. Newton’s F = ma comes out of it, and so does E = γmc². Is it all right for me to use that model?”
Your teacher will most likely say: “Mass is a measure of inertia; that is the definition.” The textbook says exactly that: “a body’s mass is a measure of its inertia, as we explained in Newton’s First Law” (OpenStax, University Physics vol. 1, 5.3). And the relativity chapter says mass is a form of energy: “Rest energy of an object is E₀ = mc²” and “we now consider mass to be a form of energy rather than something separate” (OpenStax, University Physics vol. 3, 5.9). Einstein said it first, in 1905: “The mass of a body is a measure of its energy-content” (Einstein, Annalen der Physik 18, 639, 1905).
All of that is correct, and the page agrees with every line of it. But notice what those sentences do: they say how mass behaves (it resists acceleration) and what it is equivalent to (energy). They do not say what a kilogram is made of. So:
You: “I know inertia, F = ma, and E = mc². Do any of those say what mass is made of, or only how it behaves?”
Your teacher may then bring up the Higgs: “Particles get their mass by interacting with the Higgs field; they do not have a mass of their own” (CERN, The Higgs boson). That is true for the elementary particles. But it is worth knowing the rest of the sentence: “most of the mass in composite particles, like protons, nuclei, and atoms, does not come from the Higgs mechanism, but from the binding energy that holds these particles together” (APS Physics, “Nobel Prize: Why Particles Have Mass”). So almost all of your own mass is binding energy, and the Higgs sets the small rest.
You: “So the Higgs sets the electron’s mass, and binding energy makes most of a proton’s. Is either of those an account of what the mass is, or of where it comes from? If mass is just a starting point in the course, is it OK to use the loop picture as my starting point, as long as I still get the same equations?”
You: “On the test, is there a definition of mass I should write, or only the formulas that use it?”
Read more: Mathematical Bridge, section 3 (the closed loop and the missing volume); Bridge Math Appendix, Mass and Gravity, sections 1 to 4; Mechanics Math Appendix, section 3 (E = γmc²).
You have worked through the derivation on the page: a closed null path with a minimal-area action gives m = (σₛ/c²)L_Γ and m ∝ ∮A_d ds, and E = γmc² comes back out. Here is the conversation with a physicist.
You: “I’ve worked through a derivation in which a closed null path with a minimal-area worldsheet action gives an inertial measure m = (σₛ/c²)L_Γ, equivalently m ∝ ∮A_d ds, and E = γmc² is recovered from it. The electron sets the scale through S₀ = ħ. Is there a step in that you would reject, and if so which?”
The physicist may say: “Mass is a parameter. In the Lagrangian it is the coefficient of the mass term; for the elementary particles its value is set by the Yukawa coupling to the Higgs; for hadrons most of it is QCD binding. There is nothing to derive.” That is the standard position and it is stated correctly. CERN puts it this way: “Particles get their mass by interacting with the Higgs field; they do not have a mass of their own” (CERN), and “most of the mass in composite particles, like protons, nuclei, and atoms, does not come from the Higgs mechanism, but from the binding energy that holds these particles together” (APS Physics).
What that position leaves open is the question of what carries the inertia. It is not a new question. Feynman’s chapter on electromagnetic mass ends with it unsolved: “So today, there is no known solution to this problem. We do not know how to make a consistent theory—including the quantum mechanics—which does not produce an infinity for the self-energy of an electron, or any point charge” (Feynman Lectures, vol. II, 28-5). The construction on the page is one answer: the inertia is the space a closed path hides per cycle, and the electron is a loop of finite size, ƛₑ, not a point.
You: “In the standard account the mass term is inserted and its value is set by a coupling. Is there any account, in the course or outside it, of what the object with that mass is? If a derivation exists that is consistent with every measurement and gives the same E = γmc², what would make it not a physics question?”
If the answer is “that isn’t a physics question”:
You: “Is that a theorem, or a convention about where to stop? Feynman called the electron’s self-energy an open problem. This construction has no point charge; the electron has a size, ƛₑ. Which step of it would you dispute?”
Read more: Bridge Math Appendix, Mass and Gravity, sections 1 to 4 and 6A (the worldsheet action, m = (σₛ/c²)L_Γ, elongation and observed mass increase); Particle Mechanics Math Appendix (S₀ = ħ, ƛₑ); Treatise on Caustics, Loop Closure.
You just saw a picture of why things fall. Here is how to talk to your teacher about it.
You: “I saw a picture of why things fall. Nothing pulls. Near a big mass, the lines of space lean in. A little loop of trapped light just follows the lines it lives in. Is it OK to think of falling that way?”
Your teacher will probably say: “Gravity is a force. The Earth pulls things down.” That is what most books say. NASA’s page for kids says: “Gravity is the force by which a planet or other body draws objects toward its center” (NASA Space Place, What Is Gravity?).
That is a good answer. It tells you what gravity does. But look at the next line on the same NASA page: “Albert Einstein described gravity as a curve in space that wraps around an object—such as a star or a planet.” So even NASA says there are two pictures: a pull, and a curve in space. Your picture is the second kind.
You: “How does the Earth pull on something it is not touching? Or is it a curve in space, like the NASA page says? Which picture should I use?”
If your teacher says “gravity is a force, that’s the rule,” you can say:
You: “What carries the force across the empty gap? My picture says the space itself leans. Can I keep that picture, as long as things still fall the right way?”
You: “On the test, will I be asked what gravity is, or how to calculate it?”
Read more: Bridge Narrative (step 4, the lean toward hidden space) on this site.
You have just read a page where gravity is not a pull: a loop of light standing near a mass has more space already missing on its near side, each step there costs more, and routes turn toward the costlier side, the way light bends into glass. Newton’s law comes out, and so do light bending, the Shapiro delay, gravitational redshift and Mercury’s perihelion. Here is the conversation.
You: “I’ve been looking at a model where gravity isn’t a pull. A loop of light standing near a mass has more space already missing on its near side, each step there costs more, and routes turn toward the costlier side, like light bending into glass. Newton’s F = GMm/r² comes out of it, and so do light bending, the Shapiro delay, redshift and Mercury’s perihelion. Is it all right for me to use that?”
Your teacher will most likely give you Newton, then Einstein. Newton: “a force exists between any two objects, whose magnitude is given by the product of the two masses divided by the square of the distance between them” (OpenStax, University Physics vol. 1, 13.1). Einstein: “The presence of mass—or energy, since relativity does not distinguish between the two—distorts or curves space and time, or space-time, around it” and “Gravitation is not a force between two objects but is the result of each object responding to the effect that the other has on the space-time surrounding it” (OpenStax, University Physics vol. 1, 13.7).
Both are correct, and the textbook is careful to say how they fit: “For weak gravitational fields, the results of general relativity do not differ significantly from Newton’s law of gravitation” (same page). Newton is the weak-field limit of Einstein. The model on the page gives the same Newton limit and the same four classic tests. So the question is not which equation; it is what produces the motion.
You: “How are Newton’s force and Einstein’s curvature related, and does either say what physically produces the motion?”
Here it helps to know that the best physicists have said plainly that the standard account describes and does not explain. Newton himself: “I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses” (Principia, General Scholium, Cohen and Whitman translation). And Feynman, two and a half centuries later: “All we have done is to describe how the earth moves around the sun, but we have not said what makes it go.” “Newton made no hypotheses about this; he was satisfied to find what it did without getting into the machinery of it. No one has since given any machinery” (Feynman Lectures, vol. I, 7-7, “What is gravity?”).
If your teacher says “it’s curved spacetime, that is the explanation”:
You: “What is it that follows the curve, and why does it follow? The model says the loop’s own route turns because its near side costs more, the same rule that bends light into glass. Does that disagree with anything in the course, or does it fill in the machinery Feynman said nobody had given?”
You: “On the test, is it F = GMm/r² and the field g, or is there a ‘what gravity is’ question?”
Read more: Mechanics Narrative and Mechanics Math Walk-Through (the cost map and the four tests); Bridge Narrative, steps 3 and 4.
You have followed the derivation: the same closed loop that gives inertial mass carries a stress–energy on its worldsheet, that is coupled to the Einstein–Hilbert term, and the Newtonian limit comes out with κ fixed by comparison. Here is the conversation with a physicist.
You: “I’ve followed a derivation where the closed loop that gives inertial mass also gives a stress–energy tensor on its worldsheet; that Tμν is coupled through the Einstein–Hilbert term, Gμν = κTμν, and the Newtonian limit comes out with κ = 8πG/c⁴ fixed by comparison. The relativity part is standard. What is new is where the Tμν comes from. Which step would you dispute?”
The physicist may say: “General relativity does not need a source model. You give it a Tμν, it gives you the geometry. Where the Tμν comes from is the business of whatever matter theory you couple in.” That is correct, and it is the whole point: in the standard account the Tμν of matter is supplied from outside, and the inertial mass in it is a parameter of that outside theory. Feynman’s summary still stands: “No machinery has ever been invented that ‘explains’ gravity without also predicting some other phenomenon that does not exist” and “There is no explanation of gravitation in terms of other forces at the present time” (Feynman Lectures, vol. I, 7-7).
You: “In the standard treatment, what structure supplies the Tμν assigned to matter, and why does inertial mass appear in it? Is that derived anywhere, or taken as given?”
If the answer is “GR doesn’t need a source model”:
You: “Agreed, it doesn’t need one. This construction offers one: the same loop that gives m gives Tμν, so the mass that gravitates and the mass that resists acceleration are one quantity by construction, not by a separate equivalence principle. It reproduces the Newton limit and the four tests, and it predicts no phenomenon that does not exist. If a source model exists and gives the same equations, is there a reason to refuse it, or is Feynman’s remark simply out of date?”
Read more: Bridge Math Appendix, Mass and Gravity, sections 5 to 8 (the worldsheet Tμν, the EH coupling, the geodesic, the weak-field limit); Mechanics Math Appendix, sections 4 to 8 (the four tests, worked).
You just saw a picture of what charge is. Here is how to talk to your teacher about it.
You: “I saw a picture of what charge is. A trapped photon can go around its loop one way or the other way. One way is plus and the other way is minus. Is it OK to think of charge like that?”
Your teacher will probably say: “There are two kinds of charge, positive and negative. Like charges push apart and opposite charges pull together.” That is what the books say: “Like charges repel each other, and unlike charges attract each other” (OpenStax, Physics, 18.1).
That is right. But look at how the book picks the names: “By convention, we call one type of charge positive and the other type negative” (same page). “By convention” means somebody chose the names. It does not say what makes the two kinds different. So you can ask:
You: “What is the difference between plus and minus, besides the name? My picture says it is which way the loop turns. Is there another picture of that?”
If your teacher says “electrons are negative and protons are positive, that is just how it is”:
You: “Can I use the turning picture? A loop can only turn two ways, so there can only be two kinds. That is why there is no third kind of charge. Does the book say why there are only two?”
You: “On the test, is there a picture of charge, or just the two signs?”
Read more: Electromagnetism Narrative (What is charge?) on this site.
You have just read a page where charge is the orientation of a closed loop on a shared plane: which way it faces gives the sign, the number of aligned turns per cycle gives the size (q = n·q₀), and it is conserved because that count only changes at a discrete flip or a merge or split, never a little at a time while the loop holds. Coulomb’s law and the point-charge field come out in the usual letters.
You: “I’ve been looking at a model where charge is the orientation of a closed loop on a shared plane. The facing gives the sign, the number of aligned turns per cycle gives the size, q = n·q₀, and it is conserved because the count only changes at a discrete flip or a merge or split, never continuously while the loop holds. Coulomb’s law comes out in the usual letters. Is it all right to use that?”
Your teacher will most likely say: “Charge is a fundamental property of matter. It comes in two signs and in whole units of e.” The textbook says the same: “electric charge comes in discrete amounts, and there is a smallest possible amount of charge that an object can have” and “If two interacting objects carry the same sign of charge, the force is repulsive; if the charges are of opposite sign, the force is attractive” (OpenStax, University Physics vol. 2, 5.1). The unit is known so well that since 2019 it is defined to be exact: “elementary charge 1.602 176 634 e-19 (exact) C” (NIST, CODATA 2022).
All correct. Now notice what the textbook does not do: it does not say why there are exactly two signs, or why charge comes in whole units. It lists those as properties. The word “fundamental” in your teacher’s answer means “this is where we start”, not “this is explained”.
You: “The course says charge is fundamental and comes in units of e with two signs. Does ‘fundamental’ mean explained, or does it mean ‘we start here’? Does the book say anywhere why there are two signs and not three, or why the unit is a whole number?”
It is worth knowing that this is a real open question, not a gap in your course. Dirac wrote in 1948: “The quantization of electricity is one of the most fundamental and striking features of atomic physics, and there seems to be no explanation for it apart from the theory of poles” (Dirac, Phys. Rev. 74, 817, 1948, as quoted in a 2024 review of monopole theory). “The theory of poles” means magnetic monopoles, and no monopole has ever been found.
If your teacher says “that’s as deep as it goes”:
You: “Then is it all right to start one step earlier? In the model, two signs because a loop faces one of two ways, and whole units because turns are counted. Everything the course says still comes out. What would I lose by starting there?”
You: “On the test, will I be asked what charge is, or only to use q?”
Read more: Electromagnetism Narrative (What is charge? sign, size and steps); Mathematical Bridge, section 4 (orientation and polarity); Particle Mechanics Narrative, section 7 (q = n·q₀).
You have worked through the construction on the page. Here is the conversation with a physicist. It starts with the part any physicist will recognise.
You: “I’ve been working through a construction of charge. It uses the same equations you teach, written in the language of forms: the field comes from a potential, K = dA; that makes dK = 0 automatic, which is two of Maxwell’s equations; d⋆K = J is the other two, the ones with sources; and Q = ∮⋆K is Gauss’s law, charge as the flux through a closed surface. Is there anything in that you would reject?”
The physicist will most likely say something like “No, that is just Maxwell’s equations in modern notation. Nothing new there.” That is correct. So you go on to the part that is new.
You: “Right. The new part is where the charge comes from. In this construction a charge is a closed loop of light lying on one preferred plane. Which way the loop turns on that plane sets the sign. How many aligned turns it makes per cycle sets the size, a whole number times one unit. So the sign is binary because a loop can only turn one of two ways, and the charge is quantized because a count is a whole number. That is the step I want you to look at: a closed loop producing the source J in d⋆K = J, with that sign and that unit. Is that step wrong?”
Here the physicist may say “That is an assertion, not a derivation.” That is a fair thing to say, and the answer is that the derivation is on the page: panels 6 to 8 build the source from the transported display area of the loop, and the sign from which side of the plane the loop faces. Point there rather than argue.
Then ask the question the textbook does not answer.
You: “In the course, where does the U(1) come from, and why does charge come in whole units of e?”
The honest textbook answer is that both are assumed. The U(1) is the starting point of the gauge principle: require that the phase of the electron’s wave function can be chosen independently at every point, and the electromagnetic field is what you must add to make the equations survive that requirement. It is a beautiful principle, but it is where the account begins, not something it derives. And charge quantization is not derived in the standard theory at all. A 2018 paper says it plainly: “The quantization of the electrical charge in the electrodynamics and of the hypercharge in the standard model are imposed in the theory based not on theoretical arguments but on the experimental observations” (R. Jora, arXiv:1805.09526, abstract). The one known way to derive it lies outside the course. Dirac showed in 1931 that if a single magnetic pole existed anywhere, every electric charge would have to be a whole multiple of one unit, and wrote of the idea: “Under these circumstances one would be surprised if Nature had made no use of it” (Dirac, Proc. Roy. Soc. A 133, 60, 1931, as quoted in a 2024 review). In 1948 he put it more strongly: “The quantization of electricity is one of the most fundamental and striking features of atomic physics, and there seems to be no explanation for it apart from the theory of poles” (Dirac, Phys. Rev. 74, 817, 1948, same review). No magnetic pole has ever been found. So the physicist may fairly say “we assume it; it matches experiment to many decimal places”, and that is the true state of the textbook.
You can accept that and still ask the real question.
You: “Then here is what I am trying to understand. The standard theory and this construction give the same Maxwell equations and the same conservation law. The difference is that the standard theory takes the sign and the whole-number unit as given, and this construction says why: the sign is which way a loop turns, the unit is how many turns it makes. If two accounts agree on every equation and one of them also answers those two questions, is ‘there is no deeper what’ a result of physics, or a choice about where to stop?”
Nobody is wrong here. The equations are the same. The question is only whether “what charge is” has an answer, and you now know exactly which two blanks the textbook leaves and where the page fills them in.
Read more: Electromagnetism Math Appendix (Electromagnetism from Void Transport; the Source Extension); Electromagnetism Math Walk-Through (the Maxwell set and Coulomb); Mathematical Bridge, section 4; Particle Mechanics Math Appendix (the closure phase; the dictionary labels U(1), SU(2), SU(3)).
You just saw a picture of why charges pull and push. Here is how to talk to your teacher about it.
You: “I saw a picture of why charges pull and push. One loop makes waves in the space around it. The other loop surfs on them. The side nearest the big loop always rides a bigger wave. If the loops spin the other way from each other they pull together. If they spin the same way they push apart. Is it OK to think of it that way?”
Your teacher will probably say: “Opposites attract and likes repel. That’s the rule.” The book says: “Like charges repel each other, and unlike charges attract each other” (OpenStax, Physics, 18.1).
That is right. But it is a rule, not a reason. It tells you which way, not why. Your teacher may add: “The charge makes an electric field, and the field reaches the other charge.” The book says: “The force field carries the force to another object (called a test object) some distance away” (OpenStax, College Physics, 18.4).
You: “What is the field made of? What reaches across the empty gap? My picture says it is a wave in space that the other loop surfs. Is there another picture of that?”
If your teacher says “the rule is enough for now”:
You: “Is the rule the reason, or just a way to remember which way it goes? Can I keep the surfing picture for the reason, as long as I get the rule right?”
You: “On the test, is it the rule, or is there a why?”
Read more: Electromagnetism Narrative, section 1 (the surfer) on this site.
You have just read a page where the electric field around a charge is the pattern a closed loop leaves in the space around it, its share of missing space fading as the inverse square, and the second loop’s near side always sits on the stronger part. The sign is the two facings, and Coulomb’s law comes out with kₑ as a lock.
You: “I’ve been looking at a model where the electric field around a charge is the pattern a closed loop leaves in the space around it, its share of missing space fading as the inverse square, and the second loop’s near side always sits on the stronger part. The sign is the two facings, and Coulomb’s law comes out with kₑ as a lock. Is it all right for me to use that?”
Your teacher will most likely say: “The charge creates an electric field, and the field exerts the force on the other charge. That is how the force acts across empty space.” The textbook introduces the field for exactly that reason: “Action at a distance is a force between objects that are not close enough for their atoms to ‘touch.’” and “A field is a way of conceptualizing and mapping the force that surrounds any object and acts on another object at a distance without apparent physical connection” (OpenStax, College Physics, 18.4).
Correct, and the model on the page uses the same field, the same 1/r² and the same Coulomb constant. Read that textbook sentence again, though: the field is “a way of conceptualizing and mapping the force”. Feynman said the same thing more bluntly: “A ‘field’ is any physical quantity which takes on different values at different points in space”, and of the ways of picturing it, “The most correct is also the most abstract: we simply consider the fields as mathematical functions of position and time” (Feynman Lectures, vol. II, 1-2). He added, in section 1-5: “The best way is to use the abstract field idea. That it is abstract is unfortunate, but necessary.”
You: “The course introduces the field to carry the force across empty space, and the book calls it a way of conceptualizing the force. Feynman calls it a mathematical function of position and time. So what is the field itself made of? Is that known, or is ‘the field’ the name for the place where the explanation stops?”
If your teacher says “the field is fundamental”:
You: “Is that another ‘we start here’? The model says the field is the pattern of missing space a loop leaves around it, and the other loop’s near side rides the stronger part. Then can I start one step earlier, as long as F = kq₁q₂/r² still comes out?”
You: “On the test, is it Coulomb’s law and field lines, or a question about what the field is?”
Read more: Electromagnetism Narrative, sections 1 and 5 (the surfer; the Maxwell limit); Electromagnetism Math Walk-Through, section 2 (Coulomb); Mathematical Bridge, section 4 (the sign rule).
You have followed the derivation: d⋆K = J with the retarded solution, taken to the static limit; the Lorentz force from the minimal coupling; and only at the end the names Ampère, Biot–Savart and the full Maxwell set. The sign comes from orientation. Here is the conversation with a physicist.
You: “I’ve followed a derivation that takes d⋆K = J and its retarded solution to the static limit, gets the Lorentz force from the minimal coupling, and only at the end names Ampère, Biot–Savart and the full Maxwell set. The sign of the force comes from the relative orientation of two loops. The radiated amplitude goes as 1/r and its intensity as 1/r²; the static field and the Coulomb force go as 1/r² through the source equation; the page keeps those two statements apart. Which step would you dispute?”
The physicist may say: “In quantum electrodynamics the force between charges is calculated as photon exchange. The propagator and the charge couplings give the sign and the strength, and QED is the most precisely tested theory there is.” That is correct. CERN describes it in one line: “Three of the fundamental forces result from the exchange of force-carrier particles, which belong to a broader group called ‘bosons’”, and “the electromagnetic force is carried by the ‘photon’” (CERN, The Standard Model).
You: “In perturbative QED the force comes from photon exchange, the propagator and the charge couplings. Is that exchange picture a mechanism, or is it the way the amplitude is computed? If it is a mechanism, what is exchanged in the static limit, where nothing is radiated? And where would the two accounts first give different numbers?”
If the answer is “QED is the most precisely tested theory there is”:
You: “Agreed, and this construction is aimed at QED’s static limit, not its loop corrections; it changes no tested number there. Feynman himself said the field description is ‘abstract’ and ‘unfortunate, but necessary’ (Feynman Lectures, vol. II, 1-5). The question is only whether that static limit has a geometric origin: two loops, their facings, and the pattern of missing space between them. Is there a measurement that would say no?”
Read more: Electromagnetism Math Appendix (Source Extension, sections 10 to 14; Transport Circulation from Moving Sources); Electromagnetism Math Walk-Through (Boxes 1 and 2, the Maxwell set); Electromagnetism Student Workbook.
You just saw a picture of how one photon moves. Here is how to talk to your teacher about it.
You: “I saw a picture of how one photon moves. It goes straight and hides some space behind it. When it meets a pinch in space it has a few ways to go. Where it can end up is the wave shape. It is still one photon the whole way. Is it OK to think of it that way?”
Your teacher will probably say: “Light is a wave. It is also a particle. It is both.” Books at every level say this. One says: “Under some experimental conditions, a particle appears to act as a particle, and under different experimental conditions, a particle appears to act a wave” (OpenStax, University Physics vol. 3, 6.6).
That is what the best books say. But “it is both” is two pictures, not one. So you can ask:
You: “Are those two pictures or one picture? My picture is one photon with a few ways to go, and the wave is the shape of where it can end up. Is there a different picture of what one photon does between here and there?”
It is fine if your teacher says nobody knows for sure. A very famous physicist, Richard Feynman, said the same thing about this exact experiment: “We cannot explain the mystery in the sense of ‘explaining’ how it works. We will tell you how it works” (Feynman Lectures, vol. I, 37-1). And Einstein, who discovered the photon, wrote near the end of his life: “All the fifty years of conscious brooding have brought me no closer to the answer to the question, ‘What are light quanta?’” (letter to Michele Besso, 1951). So it is a good question to keep asking.
You: “On the test, will I be asked what light is, or how it behaves?”
Read more: the Treatise on Caustics (the pinches) and the Particle Mechanics Narrative on this site.
You have just read a page where a photon’s route splits at a caustic into a small set of options with computed shares, and the familiar wave form, with ħ in it, is what the added routes give. The wave comes out; wave–particle duality is not needed.
You: “I’ve been looking at a model where a photon’s route splits at a caustic into a small set of options with computed shares, and the familiar wave form, with ħ in it, is what the added routes give. The wave comes out; wave–particle duality isn’t needed as a separate idea. Is it all right for me to use that?”
Your teacher will most likely say: “Light has wave–particle duality. In the double slit, even one photon at a time builds up the interference pattern, and you cannot say which slit it went through.” The textbook agrees on every point: “Even when electrons pass through the slits individually…interference fringes are formed gradually”, and it says why the two pictures are kept: “The wave-particle dual nature of matter particles and of radiation is a declaration of our inability to describe physical reality within one unified classical theory” (OpenStax, University Physics vol. 3, 6.6). Feynman, teaching the same experiment: “It is not true that the electrons go either through hole 1 or hole 2” (Feynman Lectures, vol. I, 37-5), and of the whole thing: “We cannot explain the mystery in the sense of ‘explaining’ how it works. We will tell you how it works” (37-1).
All of that is correct, and the model on the page gives the same fringes with the same ħ. Read the textbook’s own words once more, though: duality is “a declaration of our inability”, and Feynman says the formalism tells how, not why. Those are honest statements that the standard account calculates the pattern without giving the photon a history.
You: “Does the standard formalism give the photon a definite route between emission and detection, or does it compute amplitudes and outcomes without one? The book calls duality a declaration of our inability. Is that a fact about light, or about the theory?”
If your teacher says “that’s just quantum mechanics; nobody understands it”:
You: “Then is it OK to use a picture that gives the same interference pattern and also says what the photon did? Feynman said the formalism tells how it works but cannot explain it. Is an explanation that gets the same numbers forbidden, or just new?”
You: “On the test, is it the wave equation and the pattern, or a question about what the photon is?”
Read more: Treatise on Caustics; Particle Mechanics Narrative; Particle Mechanics Math Appendix (S₀ = ħ, the step ℓ = S₀/Ad).
You have worked through the construction: the same sum over routes with the same phase 2πS/S₀, reproducing the fold and Airy behaviour at a caustic and recovering the standard wave form, but with a finite, caustic-selected set of routes rather than all paths, and S₀ = ħ stated as a calibration lock. Here is the conversation with a physicist.
You: “I’ve worked through a construction that uses the same sum over routes with the same phase 2πS/S₀, reproduces the fold and Airy behaviour at a caustic, and recovers the standard wave form, but with a finite, caustic-selected set of routes rather than all paths, and with S₀ = ħ stated as a calibration lock. Which step would you reject?”
The physicist may say: “The path integral sums over every path. Selecting a finite set is an approximation at best, and ‘which path’ is not a well-posed question in quantum mechanics.” Both halves are the standard view and both are stated correctly. Feynman, whose formalism this is, wrote of the two-hole experiment: “It is not true that the electrons go either through hole 1 or hole 2” and, of the whole subject, “We cannot explain the mystery in the sense of ‘explaining’ how it works. We will tell you how it works” (Feynman Lectures, vol. I, 37-5 and 37-1).
You: “Is the path integral’s ‘every path’ a physical statement or a mathematical completion? Away from a caustic the stationary-phase routes dominate and the rest cancel; at a caustic the fold and Airy forms are exactly what a small set of coalescing routes gives. If a finite set with the same phase rule gives the same pattern in the stated caustic construction, what experiment separates the two?”
If the answer is “which path is not a well-posed question”:
You: “Is that a theorem, or the formalism’s silence on a question it was not built to answer? Einstein, fifty years after the photon, wrote that he was no closer to ‘What are light quanta?’ and that ‘every rascal thinks he knows the answer’ (letter to Besso, 1951). If a construction makes the question well posed, gets the same fringes, and changes no prediction in the cases it covers, what is lost by allowing it, and what would it take to rule it out?”
Read more: Treatise on Caustics (the fold and Airy asymptotics, the route selection); Particle Mechanics Math Appendix (the phase 2πS/S₀, S₀ = ħ as lock); Particle Mechanics Math Walk-Through.
You just saw a picture of where matter comes from. Here is how to talk to your teacher about it.
You: “I saw a picture of where matter comes from. A big photon hits a big squeeze and can’t fit. So it splits in two. Each half curls up and its head reaches its tail, so it gets trapped. The two halves face opposite ways. That is a pair: matter and its opposite twin. Is it OK to think of it that way?”
Your teacher will probably say: “Everything is made of atoms. Atoms are made of protons, neutrons and electrons.” That is true, and it is a good answer.
But it is a list of parts. It does not say where the parts came from. And there is something the list leaves out: every part has an opposite twin. The book for older students says: “An antiparticle has the same mass and lifetime as its associated particle, and the opposite sign of electric charge” (OpenStax, University Physics vol. 3, 11.1). And it says light really can turn into a pair: “a single photon can produce an electron-positron pair in the field of a nucleus, a process called pair production” (same page). That is the picture you saw.
You: “Where did the first electron come from? Why does every part have an opposite twin? My picture says light splits and the two halves close up facing opposite ways. Is there another picture of that?”
If your teacher says “that is university physics”:
You: “Can I keep my picture until then? It says why there are always two, and why they are opposite.”
You: “On the test, is there a picture of where matter comes from, or just the list of parts?”
Read more: Treatise on Caustics, Loop Closure (the split and the closing) on this site.
You have just read a page about pair production: a photon that cannot fit at a caustic splits, the split inflates the display area by 2⅓, that inflation is what lets each half close, and the two halves face opposite ways, so the pair and charge conservation are results rather than rules.
You: “I’ve been looking at a model of pair production. A photon that can’t fit at a caustic splits; the split inflates the display area by 2⅓; that inflation is what lets each half close; and the two halves face opposite ways, so charge conservation and the pair are results. Is it all right for me to use that?”
Your teacher will most likely say: “Pair production is real; a photon with enough energy becomes an electron and a positron. E = mc² tells you the energy needed, and conservation of charge tells you they come in opposite pairs.” The textbook says: “a single photon can produce an electron-positron pair in the field of a nucleus, a process called pair production” and “An antiparticle has the same mass and lifetime as its associated particle, and the opposite sign of electric charge” (OpenStax, University Physics vol. 3, 11.1). And “we now consider mass to be a form of energy rather than something separate” (OpenStax, University Physics vol. 3, 5.9).
All correct, and the model on the page obeys every one of those lines. But look at what they are: E = mc² says how much energy is needed, and conservation of charge says what is allowed. Neither says how a photon becomes two particles, or why the two must face opposite ways. Feynman is candid about how far the energy idea goes: “It is important to realize that in physics today, we have no knowledge of what energy is” (Feynman Lectures, vol. I, 4-1).
You: “E = mc² and conservation of charge say how much and what is allowed. Do they say how it happens? What is the in-between, when the photon has stopped being a photon and the pair has not yet appeared?”
If your teacher says “the mechanism is quantum field theory”:
You: “Then is it OK to use a picture of the in-between, as long as E = mc² and charge conservation still hold? In the model they are not assumed; they come out, because the two halves have the same closed length and opposite facings.”
You: “On the test, is it E = mc² and the conservation laws, or a question about how?”
Read more: Treatise on Caustics, Loop Closure (the equal split and the 2⅓ expansion); Mathematical Bridge, section 4 (orientation, preserved under expansion).
You have followed the construction: the equal split at a critical caustic, Fsplit = 2⅓, the maximum of a⅔ + (1 − a)⅔, lets each half close, and closure with opposite orientation gives a particle and its antiparticle with q = ±q₀. You also know where it stops short: the existence of the particular n = 1 closed route is named, not proved. Here is the conversation with a physicist.
You: “I’ve followed a construction where the equal split at a critical caustic, with display-area factor 2⅓, the maximum of a⅔ + (1 − a)⅔, lets each half close, and closure with opposite orientation gives a particle and its antiparticle with q = ±q₀. I know where it stops short: the existence of the particular n = 1 closed route is named, not proved. Beyond that, which step would you dispute?”
The physicist may say: “In field theory pair production is a matrix element. The Dirac equation requires antiparticles, and that is the explanation: there is no mechanism below the amplitude.” That is the standard account and it is correct as far as it goes. The textbook version: “Two photons can annihilate each other to produce an electron and positron pair. Or, a single photon can produce an electron-positron pair in the field of a nucleus, a process called pair production” (OpenStax, University Physics vol. 3, 11.1). And it is worth remembering how the physicist’s own field describes its open questions. CERN lists among what the Standard Model does not answer: “What happened to the antimatter after the big bang” (CERN, The Standard Model).
You: “In field theory the pair is a matrix element and the Dirac equation requires antiparticles. Is ‘no mechanism below the amplitude’ a theorem, or a habit of the formalism? The amplitude gives the rate; it does not say what happens between the photon and the pair, and it does not say why the two must face opposite ways except by conservation, which is a bookkeeping rule rather than a cause.”
If the answer is “amplitudes are all there is”:
You: “If a split-and-close construction reproduces a pair with opposite charges and equal masses, with charge conservation as a result rather than a rule, what would it take to rule it out? And is the missing n = 1 existence proof the step you would attack, or something earlier?”
Read more: Treatise on Caustics, Loop Closure (the split, the 2⅓ factor, the harmonic closure); Mathematical Bridge, section 4 (orientation and polarity); Particle Mechanics Math Appendix (the closure phase).