![[Home]](http://www.bayleshanks.com/cartoonbayle.png)
noether connection between invariants and symmetries:
not generally accepted: "General relativity by contrast, is a model theory based on a geometric symmetry principle from which its dynamics can be elegantly derived. The symmetry is called general covariance or diffeomorphism invariance. It says that the dynamical equations of the gravitational field and any matter must be unchanged in form under any smooth transformation of spacetime coordinates. To understand what that means you have to think of a region of spacetime as a set of events, each one labelled by unique values of four coordinate values x,y,z, and t. The first three tell us where in space the event happened, while the fourth is time and tells us when it happened. But the choice of coordinates that are used is arbitrary, so the laws of physics should not depend on what the choice is. It follows that if any smooth mathematical function is used to map one coordinate system to any other, the equations of dynamics must transform in such a way that they look the same as they did before. This symmetry principle is a strong constraint on the possible range of equations and can be used to derive the laws of gravity almost uniquely." -- https://en.wikipedia.org/wiki/Event_symmetry#What_it_means
"The principle of general covariance works on the assumption that spacetime is smooth and continuous. Although this fits in with our normal experience, there are reasons to suspect that it may not be a suitable assumption for quantum gravity....This is where event symmetry comes in. In a discrete spacetime treated as a disordered set of events it is natural to extend the symmetry of general covariance to a discrete event symmetry in which any function mapping the set of events to itself replaces the smooth functions used in general relativity. Such a function is also called a permutation, so the principle of event symmetry states that the equations governing the laws of physics must be unchanged when transformed by any permutation of spacetime events." -- https://en.wikipedia.org/wiki/Event_symmetry#What_it_means
Start with axioms:
derive form of Lagrangian using noether's method
take limit of massive, slow things to recover classical mechanics and electromechanics
derive special relativity
use special relativity to derive magnetism from electricity. exhibit maxwell's eqns.
general relativity: don't know yet
take limit of not-very-dense things to recover classical picture
statistical mechanics, quantum field theory: don't know yet
be careful to use explicit phase notation in place of Eulerian notation -- no 'i's allowed! ("unimaginative physics")
do every step. use mathematical document structuring (theorems, lemmas, etc, with cross-referencing by eqn numbers).
draw pictures for every other page (or somesuch). the pictures should be readable by themselves. this will allow people who don't know math to get some physical intuition.
note places where stuff sounds mystical:
the fields generated by each particle (electric, gravitational, etc) extends throughout the whole universe (although, not at the same time!)
in the macro world, we are used to things being real or not real (true or not true). at the quantum level, we have to get used to things being "sorta real". if something has a 50% chance of being true, it is "half real". (do the different sorta real possibilities interact? or do they just sum without interacting?)
when we perform a measurement, there is a wavefunction collapse. this gives the connotation of a guy who is sneakily doing things until you point right at him and yell at him, at which point his hubris "collapses" and he stays in one place. but i think a better picture is that there is Mystery in objects that we do not know much about -- and what you can do is to dive into the Mystery, to let the Mystery expand out of the object and into you. now you are part of the mystery, and it is part of you. you learn things about the object, and it is no longer mysterious to you -- but you yourself, as well as the object, are now mysterious to others.
cotter once told me that chi is both a substance and a concept at the same time. perhaps there are things in physics which are also physical entities and yet concepts at the same time.
--
i know even less about this than everything else so it may be wrong.
Quantum physics seems to really care about information. Sometimes with pairs (or probably n-tuples, i dunno) of quantities you can derive an 'uncertainty principal', which says that you can't know both of the quantities to a high precision at the same time. A common example is that if you have some tiny particle, you can't tell which position it's in at any given time without bouncing another particle off of it, but if you bounce another particle off of it, you change it's momentum; it turns out that you can never simultaneously find out both position and momentum to a high degree of accuracy. This sort of thing, where quantum physics cares about who knows what about whom (and to what precision), appears to be not just a minor footnote but a fundamental principal.
The double-slit experiment shows this. If you observe the position of a particle then in the future it acts as a particle, but if you keep yourself ignorant about its position then it acts as a wave.
In fact, quantum computers only work if you don't/can't observe the intermediate internal state of the computation!
In fact, so-called quantum eraser experiments have been done which are similar to the double-slit experiment, but in which the result of the measurement of the particle position is forgotten AFTER the particle-or-wave creates or doesn't create the interference pattern; still, erasing/forgetting the measurement causes the result to come out as an interference pattern (wave), whereas not erasing it causes it to come out as a particle. (this can't be used to send messages back in time, however; i think because the result of the whole experiment in any case can't be observed until after the choice is made regarding whether to erase the measuring particle).
(todo: repeating myself:) Quantum erasure: if part A of the system observes part B, and then 'forgets' what it saw, then the quantum computer works; but if part A communicates what it saw to the observer, quantum computer does not work. However, 'forgetting' is not like what we are used to in the macro world, and does NOT include mixing up information themodynamically (i think?); you have to 'cancel out' information to forget. If you just let the particles with the information out into the air, they'll tend to mix in their information with the air by eg slightly affecting the motion of various particles in the air, and even though the information is practically unrecoverable it doesn't count as 'forgetting' (i think this may be "quantum decoherence"? see https://en.wikipedia.org/wiki/Wave_function_collapse#Quantum_decoherence ).
Quantum computers only work if you don't/can't observe the intermediate internal state of the computation!
Quantum computers compute with 'qubits' (quantum bits), an analog of bits. Like a bit, the qubit can be in state "0" or state "1". Unlike a bit, it can also be in a superposition of these states (and more interestingly, multiple qubits can be in a joint superposition, eg two qubits could be in a superposition in which 00 and 11 are equally, highly weighted but 01 and 10 are equally, lowly weighted (i think)). If you measure/observe the qubit, it will always be in either state 0 or state 1 (and doing this 'collapses its wavefunction' so that it actually IS in state 0 or 1 at that point in time, and anything else you find out about its past or future will be found to be consistent with this). The probability of it being in state 0 or state 1 probabalistically depends on its state (that is, the superposition).
Because multiple qubits can be in a joint superposition, the states implies a probability distribution over each of the possibilities, of which there are 2^n where n is the number of bits. You might think you could use this to directly store and retrieve an exponential amount of information, but you can't, because when you measure the state, it becomes whatever you measure, wiping out all of the other information from the state. When distinct qubits are in a 'joint superposition' they are said to be 'entangled' (i think); this 'entanglement' can be thought of a as a 'computational resource' akin to time and memory in classical computation. Indeed, current efforts to build quantum computing machines seem to measure their success in how many qubits they can entangle (and for how long before 'quantum decoherence' breaks the entanglement).
Note also that the measurement you observe is probabalistic. Many quantum algorithms are only probabalistic, that is, there is some (perhaps small and tunable) chance of error.
The state is more complicated than just an assignment of probability weights to each possibility. In fact, a complex number is assigned to each possibility (two real coefficients, not one). The probability you get if you measure corresponds to the magnitude of this complex number, but the phase of the complex number matters during the quantum computation (eg while no one is looking), and in fact in some algorithms the values of qubits can effectively be combined with 'constructive and destructive interference' based on whether their phase matches, or indeed, because the phase is a continuous quantity not a discrete one, upon the degree of their phase match (i think).
A single qubit can be visualized like this: https://en.wikipedia.org/wiki/Quantum_computing#/media/File:Bloch_Sphere.svg (the "Block sphere"). The top of the sphere represents value "0" and the bottom represents value "1". When in a 'pure state', the qubit can be on any point along the surface of the sphere; so it has two degrees of freedom (3-d, minus one degree of freedom due to the constraint of being on the sphere's surface). You can think of these two degrees of freedom as being the 0/1-ness (the vertical dimension/latitude/z-axis), and the phase (the longitude). There is an uncertainty principal relationship between 0/1ness and phase analogous to the one between position and momentum; the more certain you are about the 0/1-ness, the less certain about the phase, and vice versa. Note that it's possible to directly measure the 'phase' instead of the 0/1-ness of the qubit (i think). Doing so would force the phase to be completely determined, and so would 'reset' its complementary quantity, the 0/1-ness, into a completely indeterminate state (50/50 probability) (i think). You can also 'rotate your coordinate system' so to speak, and then measure a LINEAR COMBINATION of 0/1ness and phase (i think).
The classical formulation of quantum computing uses 'quantum gates' which are unitary operators (that is, linear transformations which are isometries (rotations and/or reflections)). Note that the use of only unitary operators implies that this form of quantum computation is reversible computation.
However i think in reality you could have non-unitary transformations too [1], although i think the non-unitary transformations would represent a mixture of quantum computation and measurement/observation/(wavefunction-collapse/decoherence?). A mixture of pure states is called a 'mixed state'. In the Bloch sphere representation, mixed states can be any point in the 3d interior of the Block sphere, no longer confined to the surface of the sphere. I think mixed states are often used to represent the situation when 'your' qubits are entangled with some external system that you cannot directly observe (but i'm really uncertain of this). https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes13.pdf says "So far we have been dealing with quantum states which are what are known as pure quantum states. Here pure refers to the fact that our description of the system is entirely quantum mechanical. But we saw when we were discussing our information processing machines, that there were these equally valid probabilistic machines that had their own equally valid formulation. Is there a way to include the latter within the confines of the former and in particular to mix quantum descriptions with classical descriptions? This leads us to what are known as mixed quantum states which we will discuss in this lecture. Another reason to care about mixed quantum states is to deal with the case where we have only part of a quantum system. Thus, for instance, we may have the entangle two qubit state 1/sqrt(2)(
| 00> + | 11>). Now of course we can always figure out what quantum theory predict for our half of this quantum system by just acknowledging that this is the true description of the quantum system. But often it is convenient to discuss just one half of this quantum state, i.e. we are looking for a description of one of the two qubits for this state. It is important to realize that we must always use such descriptions appropriately: just because our description of one half is different than that of the whole does not mean that the state has changed in any way! It will turn out that the appropriate way to discuss quantum systems like this is to again consider mixed quantum states." https://quantiki.org/wiki/mixed-states says "Mixing quantum states is a basic operation, by which several different preparations are combined by switching between different preparing procedures with a classical random generator. When the outcome x of the random generator occurs with probability px, and if ρx is the state prepared upon outcome x, then the overall state generated in this way is ρ = ∑xpx ρx This expression is called a convex combination, or mixture, and since the px have to be nonnegative and add up to one, we can write convex combinations with only two terms as ρ = p ρ1 + (1 − p) ρ2. That the result of a mixture is again a state is expressed by saying that the state space of a quantum system is a convex set. Its extreme points, i.e., those states that cannot be represented as a mixture of different states with positive weights, are called pure, all other states are called mixed states." |
Note that the Bloch sphere visualization doesn't quite extend to superpositions of multiple qubits; https://quantiki.org/wiki/pure-states says "In the Bloch sphere (the state space of a qubit), the pure states exactly form the surface of the sphere. For Hilbert space dimension larger than 2, however, the topological boundary (consisting of all density operators with at least one zero eigenvalue) is much larger than the set of extreme points (which have all but one eigenvalue zero)."
For some specific classes of task, quantum computers can compute the answer exponentially faster than classical computers (eg Simon's algorithm has proven exponential speedup; Shor's algorithm for discrete logarithms and integer factorization is thought to have super-polynomial speedup, but this is not yet proven because it has not yet been proven that the best classical algorithm for these problems is not itself polynomial). But they cannot compute anything uncomputable; they are just faster. Furthermore, i think it's not just as simple as having 'ghost parallelism' from superimposed possibilities; this parallelism can only be used in certain ways.
"Grover's algorithm can also be used to obtain a quadratic speed-up over a brute-force search for a class of problems known as NP-complete." -- [2]
Quantum computing is sometimes said to invalidate the "Complexity-Theoretic Church–Turing? Thesis" [3] or "efficient Church-Turing thesis" [4], which states that a probabalistic Turing machine can simulate any model of computation IN POLYNOMIAL TIME (quantum computers cannot be simulated by classical Turing machines in polynomial time).
" Consider a problem that has these four properties:
The only way to solve it is to guess answers repeatedly and check them,
The number of possible answers to check is the same as the number of inputs,
Every possible answer takes the same amount of time to check, and
There are no clues about which answers might be better: generating possibilities randomly is just as good as checking them in some special order.An example of this is a password cracker that attempts to guess the password for an encrypted file (assuming that the password has a maximum possible length).
For problems with all four properties, the time for a quantum computer to solve this will be proportional to the square root of the number of inputs. " -- https://en.wikipedia.org/wiki/Quantum_computing
" The class of problems that can be efficiently solved by quantum computers is called BQP, for "bounded error, quantum, polynomial time". Quantum computers only run probabilistic algorithms, so BQP on quantum computers is the counterpart of BPP ("bounded error, probabilistic, polynomial time") on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one half.[83] A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP.
BQP is contained in the complexity class #P (or more precisely in the associated class of decision problems P#P),[84] which is a subclass of PSPACE.
BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is not known. Both integer factorization and discrete log are in BQP. Both of these problems are NP problems suspected to be outside BPP, and hence outside P. Both are suspected to not be NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false." -- https://en.wikipedia.org/wiki/Quantum_computing
For more on quantum computing and NP, see [5] [6]
https://en.wikipedia.org/wiki/Quantum_gate
Note that since quantum computing (with unitary matrices, eg the classical/currently standard model) is reversible, each of these gates is reversible. Some of these which are classical gates (eg Toffoli and Fredkin) are also used in reversible computing.
One qubit gates:
"One application of the Hadamard gate to either a 0 or 1 qubit will produce a quantum state that, if observed, will be a 0 or 1 with equal probability (as seen in the first two operations). This is exactly like flipping a fair coin in the standard probabilistic model of computation. However, if the Hadamard gate is applied twice in succession (as is effectively being done in the last two operations), then the final state is always the same as the initial state....Many quantum algorithms use the Hadamard transform as an initial step, since it maps n qubits initialized with
| 0 \rangle to a superposition of all 2n orthogonal states in the | 0 \rangle , | 1 \rangle basis with equal weight." -- [7] |
NOT gate: Same as the classical computing NOT gate. Also called "Pauli-X gate".
Pauli-Y gate: NOT gate and also flips the phase (NOT applied to both the 0/1-ness and the phase)
Pauli-Z gate: flips the phase (NOT applied to the phase)
Phase shift gates: Takes a rotation parameter. Rotates the phase by the rotation parameter and does not affect the 0/1ness. The Pauli-Z gate is a special case of this where the rotation parameter is pi (180 degrees).
Two qubit gates:
Square root of Swap gate: half of a swap (?). "...any quantum many qubit gate can be constructed from only sqrt(swap) and single qubit gates"
Controlled gates: eg the controlled-NOT (CNOT) gate performs a NOT operation on the 2nd qubit iff the first qubit is 1.
Three qubit gates:
CCNOT (Toffoli) gate: Same as the classical computing CCNOT gate: if the first two bits are both 1, it applies a NOT to the third bit, else it does nothing (outputs are the same as the inputs). Universal for classical computation.
CSWAP (Fredkin) gate: Same as the classical computing CSWAP gate: if the first bit is zero, does nothing (outputs are the same as the inputs); if the first bit is 1, swaps the final two bits. Universal for classical computation.
Deutsch gate: Takes a rotation parameter r. Does nothing except when the first two qubits are both 1. When they are, it returns icos(r)
| a,b,c> + sin(r) | a,b,1-c>, where r is the rotation parameter and a,b,c are the three input qubits. |
Universality:
"One simple set of two-qubit universal quantum gates is the Hadamard gate (H), the \pi/8 gate R(\pi / 4), and the ((CNOT)) gate.
A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate D(\theta)"
-- https://en.wikipedia.org/wiki/Quantum_gate#Universal_quantum_gates
"The universal classical logic gate, the Toffoli gate, is reducible to the Deutsch gate, D(pi/2), thus showing that all classical logic operations can be performed on a universal quantum computer. "
-- https://en.wikipedia.org/wiki/Quantum_gate#Universal_quantum_gates
See https://en.wikipedia.org/wiki/Quantum_algorithm
Quantum computing uses bra-ket notation which means that a value of eg 101 for 3 qubits is written
| 101> (it means more than that, but i won't go into it right now). |
---
physics heuristics (should this be in the math section?):
---
---
https://ciechanow.ski/cameras-and-lenses/
https://ahelwer.ca/post/2020-12-06-sum-over-paths/
https://physicstravelguide.com/
---
general advice on learning physics, this post seems correct in my opinion:
"I thought I would also add my two cents, though there have been many excellent responses already. I recently defended my PhD? in Physics (MIT '18).
First of all - great idea! It is never too late to learn math and physics! In fact, with hard work and commitment, anybody can muster them to a high level.
(1) Reading =/= understanding in math and physics. You understand a topic only if you can solve the problems.
(2) Work through the solved problems you encounter in textbooks carefully.
(3) Most people around me have never read any physics textbook cover to cover. E.g. reading Halliday, Resnick & Walker completely might take you years! Not all topics are equally important. Focus on the important parts.
(4) You need guidance on what is important and what is not. Online courses, college material (especially problem sets!), teaching webpages could be a helpful guide. MIT OCW is an excellent resource, once you are ready for it.
(5) Finding someone to talk to is really useful. You will likely have questions. Cultivating some relationship that allows you to ask questions is invaluable.
(4) College courses in math and physics have a very definitive order. It is really difficult to skip any step along the way. E.g. to understand special relativity, you must first understand classical physics and electrodynamics.
(5) Be prepared that the timescales in physics are long. Often, what turns people off is that they do not get things quickly (e.g. in 15-30 minutes). If you find yourself thinking hours about seemingly simple problems, do not despair! That is normal in physics.
(6) You have to 'soak in' physics. It takes time. Initially, you might feel like you do not make a lot of progress, but the more you know, the quicker it will get. Give yourself time and be patient and persistent.
(7) Often, just writing things down helps a lot with making things stick. It is a way of developing 'muscle memory'. So try and take notes while reading. Copying out solved problems from textbooks is also a good technique.
(8) Counterintuitive: If you get completely stuck, move on! Learning often happens in non-linear ways. If you hit an insurmountable roadblock, just keep going. When you return in a few days/weeks, things will almost certainly be clearer. " -- [8]
---
this is too advanced for me to know if it's right, but it probably is:
" orbifold on March 26, 2020 [–]
There are a few themes that physics revolves around:
1. Action Principle: A lot of problems in mechanics can be boiled down to writing down the correct Lagrangian.
2. Statistical physics, this teaches you about to think in terms of "Zustandssummen" and is the starting point for deriving lots of interesting laws like black body radiation.
3. Field (Gauge) Theory, turns out you can write down and derive interesting Lagrangians for Electrodynamics, Fluid Dynamics and General Relativity as well.
3.1. Noethers Theorem and Symmetries allow you to get a unified view of conserved quantities.
4. Spinors, they are fundamental for understanding the quantum behaviour of matter
5. Path Integrals necessary to understand Feynman diagrams and Calculations in Quantum Field Theory.
6. Do the harmonic oscillator in as many different ways as possible, a lot of physics can be understood by solving the harmonic oscillator or coupled oscillators. Once you've understood why this is the case and the situations in which it isn't true, you will have understood a lot of physics. ... " -- [9]
---
some quantum physics links (near the end): https://p.migdal.pl/2016/08/15/quantum-mechanics-for-high-school-students.html
---
oh the feynman lectures books are online!
https://www.feynmanlectures.caltech.edu/
---
this looks good too: https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-physics
---
https://www.susanrigetti.com/physics
---
" wanderingmind 11 days ago
| prev | next [–] |
The best lecture series I have seen till date ( and I have seen lectures by top professors across great institutions in multiple countries) is Classical Physics by V. Balakrishnan from IIT Madras, India [1]. Only people who have thought about concepts deeply over a lifetime can deliver such truly delightful lectures. If you have an hour to spare, just listen to the first lecture [2] and it will profoundly impact your outlook on science (and physics in particular)
[1] https://archive.nptel.ac.in/courses/122/106/122106027/
[2] https://youtu.be/Q6Gw08pwhws
reply "
