Probably not but I am not basing that on anything other than the LHC seems too sophisticated for it to be weaker than solar wind
The goal of the LHC and similar accelerators is to spin particles around and round to higher and higher energies using superconducting magnetic fields while solar winds are just essentially ejected particles that don't get any extra energy like accelerated particles do
but there is a ton of radiation there, just not as energetic as the beams we can produce.
it seems a little bit strange that we call something a physical theory but then have different mathematical formalisms for that physical theory. so a physical theory is not just some subset of physical phenomena with an accompanying sufficiently accurate model, but some sort of collection of equivalent such models
perhaps this is suggestive that our math is not specialized enough to the physical theory? i'm not sure
How is that strange? A theory seeks to explain phenomena; if it is already successful at it, seeking equivalent alternative mathematical models just fleshes out the theory more, never causing trouble.
In particular, a physics theory establishes a framework with which to understand physics phenomena. There is no reason why such a framework would be tied down to just one mathematical representation.
hm well im not sure why i think it is strange to have multiple genuinely equivalent ways to describe the same thing
but to me that makes it seem like our descriptions are too course grained
or something like this
or maybe it's more like it is a redundancy of the physical theory to have multiple truly equivalent formalisms. and it is interesting to think about if this redundancy can be gotten rid of
and i was also wondering if these different formalisms are really equivalent. like hamiltonian and lagrangian are just not equivalent to my understanding. sometimes you have a lagrangian but no hamiltonian or a hamiltonian but no lagrangian or somethign to this effect
which is more what i speculate; that these formalisms really are not truly equivalent. but i would like to know more about it
does anyone know of an example of a complicated physical theory that is quite successful?
I dont think that is how it is. You brought up classical mechanics. It is provable, for a certain subset that old physics did not explore beyond, that Newtonian, Lagrangian and Hamiltonian viewpoints are mathematically equivalent. However, these are three completely different philosophies and so their extensions are very different. In particular, Newtonian can do friction and other losses well, Lagrangian can do a bit of it, Hamiltonian can do almost none. Yet, in the quantum revolution,
hm well i guess something like what you are describing is also what i am alluding to. perhaps one of the many formalisms of QM will be able to extend into some QFT while the others are found to be fundamentally incompatible. selecting one of the QM formalisms in this process as the "right" one.
But there is no other way if you want to know the mathematically rigorous version. After all, rigorous maths always takes time to establish. The front third will be just straight up painful.
I hear good things about chapter 2-8 though. I'm just on the introduction, gtg sleep and receive a math award tomorrow (it's nothing special, literally just for being one of the only math majors at my local college)
Anyway, personally I would suggest the friendlier route of Feynman lectures + Ian Ford. Feynman lectures Vol 1 has those few chapters concerning thermal physics. There is an early chapter on energy, and then the few chapters before ratchet and pawl covers why we care about Carnot engine and motivates Entropy. Ian Ford starts from Entropy onwards.
@Obliv how far back? People still make their own cheat sheets. And people used to learn a lot less.
After the mathematically much less intense introduction, then Callen would make more sense.
@SillyGoose You should read Feynman's Character of Physical Law. The very issues you are considering (and grappling with often), are straight up covered right there. For example, Feynman described 3 ways of doing physics as being equivalent in classical physics: Forces style action at a distance, Maxwellian locality in both space and time, & minimum action principle.
These are 3 proven mathematically equivalent formalisms, but as you know, they invite different generalisations, and in the quantum revolution, forces are impossible to salvage, whereas we still use the generalisation of Maxwellian, and a generalisation of action principle.
another example would be something like. Consider the vector space of $n \times n$ complex matrices over the complex field $M_n(\mathbb{C})$. Define a dynamic bilinear product over this vector space by $m(X, Y) \equiv (1-t)\alpha(X,Y) + t \beta(X, Y)$ for all $X, Y \in M_n(\mathbb{C})$ where $\alpha, \beta$ are arbitrary bilinear products.
if we suppose $t \in [0,1]$, then in particular at one limit $t \to 0$ we get an algebra defined by the bilinear product $\alpha$ and at the other limit $t \to 1$ we get an algebra defined by the bilinear product $\beta$. it is also true that at every $t$ between we also have an algebra structure
@SillyGoose once again, you present a structure without presenting its motivation
I get that unfortunately math texts have evolved in a way that seems they're pulling out definitions of mathematical structures out of thin air, but that's not what's actually going on - the definitions in the books are the ones you find because they have proven, in one way or another, to be useful, and often they are directly motivated by what you find in specific examples
mathematics isn't writing down random definitions of structures until you find one that works
in this particular case, all you have here is a family of algebras $(A_t)_{t\in\mathbb{R}}$ with the additional condition that all the $A_t$ are isomorphic as vector spaces or equivalents an algebra $A$ with a family of bilinear maps $m_t$ on it. In order to justify giving this kind of situation a special name, you would first need to argue it is common enough for such a special name to be useful
sorry, but that's a silly objection - we guess the answer more or less obviously in maths and physics all the time
"let's make an ansatz" is guessing the answer, many solutions to differential equations are effectively guessed and if you look up integrals or differentials in a giant book, as is common for more complicated stuff, what do you care if that book got it by guessing
@ACuriousMind Maybe...But still, having many proofs of the same identity is a nice thing, no :) ?
But anyway, If I start doing the calculation for $n=2$ say, is there any way of doing the calculation in which the coefficient containing the factorials become manifest?
I mean, I found this in a book: but even by guessing, could I come up with the coefficient in a "natural" way? That is how do I organize my calculation for this particular problem?
You see that's the problem...Even after doing upto $n=5$ which is tedious already, it is kinda hard to guess that form
But when you write it like that, it seems familiar from somwehere...
This is the expression (from Poisson and Will's book on Gravitation) which I want to prove ultimately though...It expresses symmetric tracefree products of vectors in terms of ordinary tensor products; it is useful for evaluating the trace term with the correct normalization
I got to (probably) the original reference where Pirani derives this, but it uses that above result which he asks the reader to derive via induction. How did he guess it!!!!
@ACuriousMind well the inspiration was thinking about how something like poisson brackets can dynamically become something like a commutator. but the aforementioned idea does not really make sense so i thought to work with a major simplification (which is what i wrote out)
i am just interested in the maths, but it seems like the general idea is loosely related to deformation quantization (though I don't know about deformation quantization)
but the idea was motivated by i think perhaps what is the same motivation for deformation quantization
@ACuriousMind i feel like this definition is not very random. it is--to me at least--the natural generalization of any algebraic structure one encounters: make the operations parameter dependent instead of static. i am surprised though that it seemingly has no valuable use if it does not
@SillyGoose Why would this be natural? When we do group theory, do we ever consider a group with such a parameter-dependent group operation? If this was "natural", you should be able to produce plenty of other examples where we do this
@SillyGoose See, that's exactly the kind of motivation I want to hear! Because for quantization the answer is the deformation quantization usually deforms the star product, not the Poisson bracket itself, and also that you need to phrase this differently (in terms of formal power series) in order for the limit $\hbar \to 0$ to make sense
well we think about $\mathbb{R}^3$ and curves through it. why don't we think about the set of all vector spaces (or an appropriately defined subset) and curves through it?
we think about paths and loops all the time, but do not talk about paths or loops in spaces of spaces
the Graßmannian is a standard construction that's the "space of all subspaces of a space", for instance, and the loop space is also a standard construction
i mean a path in the set of all vector spaces, for instance. not fix a vector space and talk about data related to it
so to speak approximately, to define a path $\varphi: [0,1] \to \textbf{Obj(Vect)}$ where $\textbf{Obj(Vect)}$ is the objects of the category of vector spaces
this is a more general version of what i wrote above but this is the idea i was interested in
well i am okay with the idea not being generally definable i guess
in the example i typed below the note it seems like something i would call as inducing a path in the set of all algebras
in particular, one might be able to restrict their attention sufficiently enough to some appropriate subset of the set of all [insert particular structure]s to have well-defined "paths"
well another angle that is perhaps something that is actually used or done is the following. is there a physical theory in which the state space is dynamic? i feel like in QFT the answer is yes but i am not certain.
like at one time you will have one Hilbert space, at another time you will have another Hilbert space according to some governing equation
in some constructions of QFT, you have an association of a Hilbert space to any spatial slice (Cauchy surface)
given a global time function, this of course gives you a "path" $t\mapsto H_t$ "of Hilbert spaces", but since all separable Hilbert spaces are abstractly isomorphic anyway it's important to remember that this is more about the operators one can talk about at each point rather than something about the space itself
@Sanjana In that particular case, each derivative of $r^{-1}$ ought to subtract from the power by one each time. However, because of the square root of the sum of squares, and that each derivative brings out a position variable in the numerator, this derivative is somewhat naturally extracting two powers each, and then putting one leftover in the numerator. Hence the numerator has $(2n)!$ and the denominator $r^{2n+1}$
In fact, what is really happening is that you have a double factorial in the numerator, which then explains both the numerator of $(2n)!$ and the denominator of $2^nn!$ in one go.
@SillyGoose For vector spaces, classification is simple - it's just $\mathbb{R}^n$ for arbitrary $n$. I don't see what this kind of classification has to do with your "paths".
I really think you're overgeneralizing here without actually looking at any concrete problem you could apply your generalization to
well i think i don't have a problem that this object is supposed to solve. it is just an interesting way to generalize a little bit of the math formalism found in classical and quantum mechanics
well it'd just be theoretical exploration :P. like what if you scale the commutator as $\frac{1}{t}[X, Y]$ so that in the limit $t \to \infty$ all observables "commute" or something like this. this is a naive idea, so i am not sure how sensible it technically is
or have something like $e^{-t}[X, Y]$ which decays exponentially
@SillyGoose As I said, deformation quantization does this usually by scaling the star product via formal power series - you don't need to invent new stuff for this, you need to just learn what's already out there
it indeed seems like such a definition would lead to at least a few technical problems
well i started off all this by asking if there was a name for what i was describing :P
and i saw from le google that deformation quantization had a loose resemblance conceptually
but i do not care so much about how to quantize a classical theory--more about the what the mathematics is that is being used (if it's being used) in the deformation quantization. in particular, what i assume is referred to as the "deformation"
i can see stduying this particular example of the mathematics as being instructive, but my original question can perhaps be restated as what is the general mathematics used to do the "deformation" in deformation quantization. (im not sure if i am referring to the correct part of deformation quantization, i am just assuming deformation is used in the colloquial sense in the naming)
You start with your algebra $A$ and then the deformation is via a star product/bracket on $A[[\hbar]]$
@SillyGoose Well, yes, but what you want to model in deformation quantization is that the commutators of observables that don't depend on $\hbar$ can contain higher powers of $\hbar$
just making a $\hbar$-dependent bracket on the algebra of observables $A$ doesn't do that
but we have $[q,p] = \mathrm{i}\hbar$, i.e. the l.h.s. is the bracket of two things constant in $\hbar$ but it produces things not-constant in $\hbar$
and besides, you want to have the observables dependent on $\hbar$ in this way, too, because we want to talk about quantum correction to classical observables, i.e. then we can look at the "corrections" that vanish from expressions as $\hbar \to 0$
that's why I keep harping on the point that you need to look at the concrete problem you want to model in detail before you go off talking about abstract generalizations - the power series approach is tailored to make it possible to talk about all the things physicists want to talk about in the context of quantization and classical limits
And to throw a spanner into the whole discussion, you definitely have heard multiple times here that $\hslash$ is a unit-ful quantity, and is thus not at all suitable as a parameter to expand power series by. All those arguments that we physicists are hand-waving about, if you want to be rigorous, and make concrete mathematical arguments thereof, you have to first find the appropriate parameter to expand the power series by, and then redo all those arguments.
Of course, physicists arent just talking total nonsense. In every case, there must be something that serves as the parameter to expand by. But it is clear that you are having a headache worthy relationship between maths and physics, so much so that ACM had had to come in and point this out multiple times.
@naturallyInconsistent ah, well, in natural units you have $\hbar = 1$ so that's not particularly worrisome: In natural units we just have a power series parameter that is "classical" at 0 and "quantum" at 1
@ACuriousMind That does not always work. For example, in QED, the actually sensible parameter to expand upon, is $\alpha$, which happens to be reciprocal of $\hslash$
and whether you're doing QED or QCD or whatever doesn't really have any influence on the basic commutation relations that you need to model, i.e. $[\phi,\pi] = \mathrm{i}\hbar \delta$ - there's always that $\hbar$ in the CCR
@ACuriousMind That I know, but I wanted to make it clear that certain expansions don't make sense. For example, if you naïvely wanted to take the $\hslash\to0$ limit of QED, that seems to take $\alpha\to\infty$, which then gives us nonsense.
@ACuriousMind Miao miao can always choose for that $\hslash$ to go into the $\pi$; you can scream in horror now
@Mr.Feynman in miao miao version of the unit system for quantum theory, there is no appearance of $\hslash$ nor $c$ nor $e$ nor a whole host of other quantities; even if there is a want to expand, it is not even there to expand from.
this is from feynman's character of physical law. i am a little skeptical of the claim being made. in particular, i don't see a conception of conservation law of angular momentum which makes the claim true/surprising (not surprising in the sense that if one constructed a physical theory to particularly encode conservation laws, it would not be surprising that such conservation laws held)
but i think this is perhaps a misunderstanding coming from me not knowing what it looks like to experimentally measure "angular momentum" of an isolated system
is this the same notion as specifying a potential by wilson loops?
or i more mean is specifying a potential by wilson loops the souped up version of the above
@SillyGoose you need to think like an experimentalist here: Angular momentum is just $r\times p$
and of course, before we know Noether's theorem that connects the conservation law to rotational symmetry, it is very surprising that this quantity is conserved in so many different contexts
@SillyGoose I'm pretty sure the point of this text is that at this point you're not supposed to already know or wonder about all the mathematical abstraction you could throw at a particular observation :P
@ACuriousMind Last time I was also having a similar doubt while conversing with nIs. I want to get it straight and clear once and for all. What is the expansion parameter in Born-Dyson series? $e$ or $\hbar$ or $\alpha$ or what?
nIs convinced me that $e$ or $\hbar$ are dimensionful so they can't be used as expansion parameters. So it is $\alpha$, but I swear I have seen people doing an expansion in the coupling and also sometimes in $\hbar$. E.g. In perturbation theory, the expansion is done in coupling; but the $\hbar$ counts the loops
(Don't read the first half of the sentence in the E.g. : it doesn't make an overall sense)
the naive Dyson series is in inverse powers of $\hbar$, actually
but there's lots of subtly different perturbation expansions in QFT depending on what you resum, this question doesn't have an unambiguous answer
also it's not unambiguous for any fixed series, either - if I have $f(x) = \sum_n \lambda^n f_n(x)$ as an "expansion in $\lambda$", you can trivially turn it into an expansion in $\lambda' = \lambda/g(x)$ for any function $g$: $f(x) = \sum_n \lambda'^n g(x)^n f_n(x)$
yes. the Dyson series solves the time dependent Schrodinger equation perturbatively, similar to how the WKB expansion solves the time independent Schrodinger equation perturbatively
choosing (dimensionful) constants as your $g(n)$ then switches between "expansions in $\hbar$" or "expansions in $\alpha$", this is the least troubling aspect of perturbation theory :P
you may insist on choosing a manifestly dimensionless and "good" expansion parameter in order to re-assure yourself the series makes physical sense, but this is not required by the mathematics
yes. mathematics doesnt care about dimensions. to mathematics, the schrodinger equation is just a differential equation after the numerical value of $\hbar$ has been fixed by a choice of units
@ACuriousMind independent of this, are the two concepts indeed related? that the average potential over a sphere is equal to the value of the potential at the center of the sphere and reconstructing a potential from whatever wilson loops are?
i recall an exercise in griffiths showing the former fact :P it would be interesting that the fact shows up again in the latter form if true
@SillyGoose no, the two are unrelated since the equivalence between connection forms and parallel transport is general while the thing with the average is due to the mean value property of harmonic functions and the potential being harmonic in this case
@ACuriousMind the authors of that paper claim there is an analogous "arrow" of time for microscopic phenomena as well, that follow from the results of their paper. (The article)
