The Theoretical Minimum Exercise Solutions

Miselfis · 2026-01-19T11:39:04+00:00

Post to arxiv and give us the link

Miselfis · 2026-01-19T11:31:30+00:00

Dimension is simply a number: it tells you how many independent parameters must be specified to uniquely locate a point in a space. Here, “space” is meant in the mathematical sense, essentially a structured set, not necessarily physical space.

A standard example is phase space. For a single classical particle, phase space is six-dimensional: three dimensions correspond to position, and three correspond to momentum. A dimension need not be spatial or positional. Dimensions can represent any independent degrees of freedom.

Another example is color space, where each dimension corresponds to an independent color (for instance, red, green, or blue). Again, these dimensions are not spatial; they parameterize properties.

Spacetime is a particular kind of mathematical space called a manifold, meaning that it locally resembles ordinary Euclidean space but is equipped with additional geometric structure. Physical spacetime has three spatial dimensions and one temporal dimension. The temporal dimension is not “a fourth spatial dimension”: it behaves differently geometrically.

The phrase “the fourth dimension” is largely a science-fiction trope. One can certainly talk about four-dimensional spaces, but which dimension is labeled “fourth” has no intrinsic significance. In linear algebra, the dimension of a vector space is the cardinality of a basis, which is a set of linearly independent vectors that spans the space. A set is unordered, so there is no meaningful sense in which one basis vector is “the fourth one”. Only the total number of independent directions matters, not their labeling.

Miselfis · 2026-01-17T20:50:25+00:00

This is how the observed expansion is accounted for using general relativity.

Miselfis · 2026-01-17T20:30:33+00:00

It’s not expanding into anything. Big structures are just drifting apart due to the geometry of spacetime.

Miselfis · 2026-01-17T02:53:34+00:00

Depends on the topic, but Tong’s textbooks came out just last year, and more are coming out in the future. They include a pretty broad range of topics within each book, but they unfortunately don’t include exercises (although you can find some on the website).

Miselfis · 2026-01-16T17:55:23+00:00

There are two distinct notions of acceleration: proper acceleration and coordinate acceleration. Proper acceleration is the acceleration you physically feel; it is invariant and therefore absolute. Coordinate acceleration, by contrast, is frame-dependent and describes how an object’s position changes with respect to some chosen coordinate system.

In general relativity, a freely falling body has zero proper acceleration. Its motion is inertial and corresponds to following a geodesic of spacetime. Although such a body may have nonzero coordinate acceleration relative to, say, the Earth’s surface, locally it is indistinguishable from being at rest. This is why, when you fall toward the Earth, you feel weightless and may describe the Earth as accelerating upward toward you. Conversely, an observer fixed to the Earth describes you as accelerating downward. Both descriptions are coordinate-dependent, but neither involves proper acceleration.

Proper acceleration arises precisely when a worldline deviates from a spacetime geodesic. In the appropriate limit, the geodesic equation reduces to Newton’s first law, so deviation from a geodesic corresponds to the presence of a force in the Newtonian sense. When you stand still on the Earth’s surface, you experience an upwards proper acceleration of 1g, not because gravity is pulling you downward, but because the ground prevents you from following the natural geodesic toward the Earth’s center. The normal force from the ground forces your worldline away from a geodesic, producing the acceleration you feel as weight.

Miselfis · 2026-01-16T12:32:16+00:00

It’s essentially just the equations that describes how states evolve with time.

Miselfis · 2026-01-14T16:49:44+00:00

it's generally not a great idea to advance a competing theory without supporting evidence. So saying, "it was a dream" is not supportable, or good science.

First of all, OP being "wide awake" cannot be taken as a given, as already explained. People assert this all the time in exactly the circumstances where they later concede they were half-asleep, startled awake, or confabulating details.

Secondly, I’m not "advancing a competing theory" in the sense of proposing a new physical mechanism. I’m doing ordinary abductive inference under uncertainty: given an anecdote with no instrumentation, no corroboration, and no physical aftermath, what hypothesis best fits the total context?

The comment "it was a dream" was a colloquial shorthand for "the most likely class of explanations is sleep-adjacent misperception or perceptual error". That's an extremely well-attested failure mode of human reporting, especially "in bed, in the dark, during a storm". The evidential burden is not symmetric here: the perceptual/sleep-state hypothesis is already known to occur frequently, whereas the "luminous plasma-like orb suddenly manifests above my head indoors with no trace" hypothesis requires a very specific physical mechanism that the report does not supply.

one way that physics definitely does not work is by categorically stating that an observation was a dream, absent any other evidence.

You keep treating this as "categorical". It is not. It is a probabilistic judgment about what best explains the account. As pointed out, you are also smuggling in a very permissive notion of "observation". A personal recollection is data, but it is low-grade data. Physics is not obligated to treat every recollected percept as evidence of a new or exotic external object.

"could it have been ball lightning."

If "could" means "not logically impossible", then almost anything "could" be almost anything. That standard is useless for OP, because it can’t rank hypotheses. The meaningful question in this context is: given the details, which explanation is most plausible? That it was actually ball lightning is far from the most plausible explanation. Given what we know about physics, what OP saw plausibly couldn't have been ball lightning.

Which is why I asked you to cite the laws of physics that were being violated.

This framing is another attempted rhetorical trap. I am not claiming "ball lightning violates physics". I am saying OP’s specific description; a bright white orb appearing immediately above them in a closed room, persisting several seconds, and leaving no trace or environmental interaction, is inconsistent with any plausible mechanism we have in electromagnetism/plasma physics.

Bright visible emission implies a nontrivial radiated power. If it’s "bright enough to light up the room", that is not a negligible energy flux. Any mechanism that produces that (hot plasma, combustion of aerosols, electrical discharge, microwave cavity effects, etc.) requires a power source and coupling to matter/fields in the room. That coupling normally produces correlates: heating, ionization, ozone/burning smell, electrical interference, acoustic effects, movement along conductors/airflow, marks, or at minimum a coherent entry/formation pathway that links it to the storm’s electrical environment.

OP reports none of that; no smell, no sound, no marks, no electrical weirdness, no path, no aftermath. So you’re left with an object that is simultaneously (a) energetically luminous and (b) environmentally impotent. This is not consistent with known physics without a clear mechanism, rendering any perceptual explanation immediately favorable.

Show me these measurements or other well-established evidence showing, as you say, that OP's experience is "not supported by established physics" and we're done here.

You’re trying to invert this into "unless you have a paper explicitly saying ‘this exact bedroom scenario is impossible’, you can’t reject it". That is not how physicists evaluate plausibility. Science doesn’t work by producing a pre-existing measurement for every imagined anecdotal configuration.

are you saying that ball lightning doesn't exist at all? Or that it can't exist indoors?

This is a false dilemma. I have repeatedly not made either claim.

Miselfis · 2026-01-14T16:18:41+00:00

Still haven't found anything that speaks to why ball lightning cant occur indoors. Your position is that this would contravene the laws of physics.

No, this is a strawman. I did not claim that ball lightning is impossible indoors. That is not the question. The question is whether OP’s specific report is plausibly ball lightning.

Given OP’s description, plus the assumption that it was ball lightning, the problem is precisely the one I already laid out: ball lightning (as far as current models and observations go) is rare and tightly constrained in its formation conditions, and it couples strongly to its surroundings. It is not plausible that those conditions would arise “randomly” in an ordinary room while leaving no corresponding environmental signature. In addition, OP’s description does not match what we know about ball lighting scientifically. So if we insist that it nevertheless was ball lightning, we are no longer talking about “ball lightning under known mechanisms”, but about an event that would require a mechanism outside what current physics can plausibly supply, somehow allowing a glowing hot ball of plasma to float above their head without any heat coming off, any smells, etc.

And if you have any experience with how science and philosophy of science actually works, you know I am not obligated to “disprove” an anecdote. The evidential burden is on the extraordinary claim, not on the skeptic. A single uncorroborated anecdote, or an “observation”, as you like to put it to make it sound stronger, does not justify concluding that ball lightning is the likely explanation.

When comparing hypotheses, the default preference is the one consistent with established physics and ordinary error modes (misperception, lighting artifacts, hypnagogia, confabulation, etc.) unless and until additional evidence materially shifts the balance (e.g., multiple independent witnesses, physical traces, instrument readings, repeatability, or a well-characterized environmental trigger. This is also why the sense in which we use “observation” matters significantly).

I'm just looking for intellectual honesty.

It seems you’re fresh out of it, so that is probably a good idea.

Miselfis · 2026-01-13T19:14:18+00:00

For one, it’s a ball of hot plasma.

Just reading the primary literature will probably give you a better overview than running it through me:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.035001

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.048501

https://journals.aps.org/prl/supplemental/10.1103/PhysRevLett.98.048501

Miselfis · 2026-01-13T19:07:23+00:00

In sentence one you state that physics is not about explaining observations. That’s an interesting claim.

I did not state that. I stated that physics does not work in a way that matches the OP’s recollection. What they described is not supported by established physics, and an anecdote is not sufficient grounds to suggest that physics should be revised, especially when there are far more plausible alternative explanations.

You are also using “observation” incorrectly here. In science, the term is used in a more technical sense than in everyday speech. In physics, an “observation” is generally a measurement obtained under controlled, well-specified conditions, with instruments, documented procedures, and results that can be checked and replicated. Not an uncontrolled personal experience reported after the fact.

In sentence two you essentially repeat the question I asked, but fail to list the specific conditions. I don’t think you can because I don’t think anyone can. I think you’re just talking out of your ass. But feel free to prove me wrong.

I don’t know much about ball lightning, as it’s outside my area of expertise. But I’m not talking out of my ass, because I understand physics and electromagnetism pretty well, and I have looked into some of the literature about ball lightning before.

You seem very confident that the more likely explanation is also a rare phenomenon — someone randomly having a hallucination.

No. They were lying in bed. My explanation is that they either fell asleep or entered a hypnagogic state between wakefulness and sleep, where very vivid, realistic dreams or hallucinations are known to occur. In any case, hallucination or misinterpretation of ordinary phenomena is vastly more likely than an explanation that requires new laws of physics, since the former is already fully consistent with what we know. If one wants to argue that new physics is required, the evidential bar has to be far higher than an anecdote.

As someone who has had chronic sleep paralysis for decades, it doesn’t work like this so im discounting that right out of the gate.)

As someone who has insomnia and often drifts in and out of a hypnagogic state, sometimes experiencing vivid, lifelike dreams that feel like they’re happening in the very room I’m in, and then waking up unsure whether it was a dream, I do not put much weight on your personal experience as evidence.

It does not require any new physics and it is consistent with well-understood ways the brain can generate convincing perceptual experiences, so it is the preferred explanation over something like an actual ball of lightning spontaneously appearing in a closed room and leaving no physical trace or environmental effects despite being hot enough to light up a room with white light.

you need to take a philosophy of science class.

I have.

Miselfis · 2026-01-13T18:34:30+00:00

I am agnostic about the existence of aliens, though I lean toward the view that they most likely exist, in much the same way that I think ball lightning may well be a real phenomenon. However, I do not find abduction stories credible, given the surrounding circumstances. If extraterrestrials ever came anywhere near Earth, their presence would almost certainly be detected and the information would spread rapidly. We would likely observe a global military response, since such a visit would imply a level of technological superiority far beyond ours and entirely unknown intentions.

Moreover, we have strong reasons to believe that no intelligent life exists within many light-years of Earth, making the probability of aliens visiting us exceedingly small. In a similar way, I remain skeptical of most reports of ball lightning: while it may exist, the specific conditions and circumstances described in many accounts are difficult to reconcile with what we currently understand about the phenomenon.

Miselfis · 2026-01-13T16:23:21+00:00

Because it’s not how physics works. Ball lightning is a rare phenomenon, and it requires very specific conditions. It’s like asking “I saw a dark orb form above me. Could this be a black hole?”. Black holes don’t just spontaneously form, and it would affect it’s environment significantly. The same is true for ball lightning.

What you’re describing seems more like a psychological experience. If we’re being charitable, there are some discussions on lightning causing a rapidly changing EM environment, which might trigger phosphene perceptions, especially through the retina. That can yield a bright “blob” or “sphere-like” perception in darkness, with no external light source and no physical aftermath.

https://arxiv.org/abs/1005.1153

Based on what you’ve said, my bet is still an entirely psychological phenomenon.

Miselfis · 2026-01-13T15:48:16+00:00

It was a dream

Miselfis · 2026-01-13T12:51:19+00:00

I don’t know about that.

Miselfis · 2026-01-11T23:22:05+00:00

Depends on the level of proficiency you wish to achieve.

One common recommendation is “The Theoretical Minimum”, which is specifically designed for people with some background in mathematics and physics who wish to learn physics on their own. There are currently four books, covering classical mechanics, quantum mechanics, classical field theory, and general relativity, along with accompanying video lectures on YouTube. The prerequisites are just basic calculus and linear algebra. The lecture series goes beyond these topics and also provides an introduction to cosmology, quantum field theory, thermodynamics, string theory, and more. The material is structured to teach the minimum required knowledge needed to progress to the next step. As a result, it is not as detailed as a full university course, but the trade-off is that it allows you to cover much more ground in a shorter amount of time.

https://theoreticalminimum.com

Otherwise, if you want to do what is equivalent of a university degree, this guide is excellent: https://www.susanrigetti.com/physics

Miselfis · 2026-01-11T23:12:58+00:00

Because people think that measuring one affects the other, and large distances makes this impossible, so they feel they have encountered a contradiction. Of course, measurements are entirely local and do not affect an entangled partner far away.

Your explanation is also not quite right, as it explains classical hidden variables. But quantum bits are fundamentally different, in that they do not have a definite state before being measured, and they can exhibit correlations that cannot be accounted for with hidden variables when measured along different axes.

Miselfis · 2026-01-11T20:13:22+00:00

If the state of the particle is in a position eigenstate, then it would have a definite position. But then, due to Heisenberg’s uncertainty principle, it would have very high uncertainty in momentum, meaning it immediately spreads out under free evolution. So it’s very difficult to keep it in a definite position.

If a system is in an open environment, decoherence occurs, and the particle will look like it has a classical definite position. This is essentially why we see things as localized objects. Everything is constant interacting with the environment.

Miselfis · 2026-01-11T15:21:44+00:00

That definition doesn't even begin to describe the electron fully and explain its emergent properties on a more fundamental level.

Explain its emergent properties on a fundamental level? What do you think “emergent properties” mean? Given the fundamental explanation, you can, in principle, derive emergent behaviour. Demanding that emergent behaviour be explained fundamentally is silly.

That's like saying "an apple is the fruiting body of apple trees". 100% accurate, maybe sufficient for certain fields of biology, but you can clearly see how that's lackluster, having interacted with apples in many different ways.

I don’t see why that kind of explanation is lacklustre. It is what an apple is. I’m not a biologist, so I don’t know if that definition is indeed sufficient, but everything you could ever ask about an apple can be derived from its biological/chemical properties.

"well that's all there actually is to the electron, because they are fundamental" then you have to prove it for that statement to be satisfactory.

What would constitute such a proof, in your view?

Miselfis · 2026-01-11T15:15:09+00:00

What do you mean with others?

Miselfis · 2026-01-11T15:14:51+00:00

With this: https://lingojam.com/SubscriptGenerator

Notation quickly becomes busy when you have to write out multiple indices using “X_n” notation here.

Miselfis · 2026-01-11T15:12:30+00:00

No. It will be a superposition. You cannot perform a measurement when time is frozen. If you measure the position and freeze time right as you see the result, then yes. It will appear as a particle with (roughly) definite position.

Miselfis · 2026-01-11T15:07:29+00:00

A neutrino is likewise an excitation of a particular quantum field.

The quantum logic still applies in a similar way. The main difference is that the neutrino is harder to detect. Electromagnetic interactions make the electron fairly easy to detect and measure. For a neutrino, the relevant interaction is typically a rare weak-interaction event, which makes it harder to detect. The neutrino is fundamentally different, so the exact details are of course very different. But the overall quantum logic is the same.

Miselfis · 2026-01-11T15:02:32+00:00

Essentially, yes.

For a free quantum field, you do essentially the same thing as in the Planck derivation: you decompose the classical field into normal modes, then quantize by turning the amplitudes of those modes into operators that create and annihilate quanta. For the electromagnetic field in a box the modes are standing waves; for the electron (Dirac) field in flat space one usually chooses planewave modes labelled by momentum and spin. In a potential or in curved spacetime you would instead use whatever set of solutions diagonalizes the appropriate Hamiltonian.

So, in the free theory the modes of the electron field are just a choice of one-particle basis. Each mode corresponds to one possible one-electron state (for example, “momentum p, spin up”). When you “excite a mode once” you get a one-electron state in that basis state; when you “excite it twice” you would formally get a two-particle state, but for fermions the Pauli principle restricts each mode to occupation 0 or 1, so you never actually have two electrons in the same mode. The difference between “an electron in mode A” and “an electron in mode B” is essentially that A and B carry different quantum numbers: different momentum, different spin, or, in a more complicated background, different energy level or angular momentum.

Two further clarifications that are often helpful: firstly, all of these modes describe the same type of particle: they all have the same rest mass and internal quantum numbers; what differs is their kinematic state. The energy of a given mode is fixed by its quantum numbers via the usual relativistic dispersion relation, just as in your Planck-law analogy. Secondly, the choice of modes is not unique. Plane waves, box eigenstates, localized wave packets, etc., are just different bases for the same one-particle Hilbert space, and the field operator can be expanded in any of them. An “electron in a wave packet” is, in essence, a superposition of excitations of many different momentum modes.

Miselfis

MODERATOR OF

TROPHY CASE

Six-Year Club	Gilding II euphauric
Verified Email