/sci/ - Science & Math

>>6396308
whats going on in that picture and at what year university do you learn about it?

>>6396328
seems like renormalization.

>>6396308
I'm a math student.
How much of classical field theory I need to understand to grasp the physics of QM? The math is not a problem currently.

>>6396328
It's the soul of physics, the renormalization group flow.
I did my masters some years ago, read about QFT for procreation currently. Made a thread here yesterday, but it died.
>>6394943
I'm, in fact, interested in general generation functions for stochastic processes at the moment.

See also
>>6396345
motivation comes from reading about the zeta function of a language here
http://en.wikipedia.org/wiki/Regular_language#The_number_of_words_in_a_regular_language
and
>>6395317

>>6396349
it's difficult to quantify, especially if you don't say to what extend you want to learn it. "Quantum mechanics" is a broad subject. So in general, given that e.g. quantum optics, qunautm computation, etc., is just as much QM, I'd say "not much".
If you want to understand the standard model you should, at the very least, read a physics (!) text on electrodynamics in the relativistic formulation.

>>6396369
Thanks for the answer. I was more worried about understanding how does one choose that kind of mathematical formulation starting from the various experiments, which as far as I know are very "electric".

>>6396374
My mantra is that while learning physics, one should always keep track of what it is that one finally wants from the theory. Having the hierarchy of relevance sketched out in front of you makes it also much clearer at which point and why physicists might gloss over formalities.

how does qft work in curved spacetime?

why do we always measure a single value for the order parameter after SSB, instead of a general superposition? I understand the thing about representations being inequivalent and having different vacua, but I just can't shake the feeling that this is just a restatement of the previous sentence.

My understanding of QM is that most things are treated as waves.(?) or fields. If this is the case what forms of approximation are there on particles? Could you not treat the fields as a 3 dimensional signal and use different approximation theory such as wavelets?

Sorry, I'm just a Math student who is really interested in physics, approximation, and DSP.

>>6397443
I don't think I know an answer or maybe not even quite understand the question. Do you mean "measuring" of the order parameter literally, physical measuring? And what do you mean by superposition?
The first thing I think about would be the hat potential and any addition structure I'd imagine on the exp(i\theta)-parametrized vacua would just result in another fixed value. Or do you maybe mean you assign e.g. a probability p and (1-p) to two configurations and weight the observables computed from the theory accordingly? You might mean something technical and I'm crucially misunderstanding you.

>>6397455
Very broad question. Given that QM is the theory giving you lasers, of course the spectral analysis, filtering of relevant signals and so on, that can be applied to the results of the theory.
If you speak of approximation to the (potentially unobservable) theory itself, maybe
http://en.wikipedia.org/wiki/Quasiparticle
is of interest.

>>6397505
Sorry, I used the word measure but regretted it as soon as I sent the comment.

I mean, very practically: if we have a point particle in a higgs-like potential, and we quantize this system canonically, the ground is degenerate and the degenerate eigenspace are invariant under the symmetry.

If instead we consider a quantum field with that potential, we get a single, nondegerate ground, asymmetric and with a particular, but presumably arbitrary v.e.v.; basically there are as many different representations as there are values for the order parameter, and these 'universes' don't communicate.

I understand how this is possible in QM_infinity, Wigner's theorem, Godstone's theorem and shit. But I don't get why it happens, what is the mechanism through which we never observe superposition between different higgs vevs.

For example, I assume you could put (with some difficulty, granted) a quantum magnet in a superposition for magnetization, below the Curie temperature. Is this correct? Why can't we do the same for a field on all spacetime?

>>6397557
>comment
you mean post

>>6397557
>>6397443
would also like to know this

>>6397557
superselection

>>6397572
idgaf
>>6397579
thanks I'm checking it out

OP here again
>>6397557
If <span class="math">|0\rangle[/spoiler] is your vacuum state and Q is the generator in question, then <span class="math">[H,Q]=0[/spoiler] impies that there is a conserved current density <span class="math">J^\mu[/spoiler]. As in the case with the electrical charge, the zero's component of the current, or in fact
<span class="math">Q=\int J^0 dV[/spoiler]
That's the given part of the story.
Now naively there are angles s so that, with
<span class="math">u(s) := \exp(i s Q) = 1 + i s Q + (1/2) (i s)^2 Q^2 + ...[/spoiler]
the state <span class="math">u(s)|0\rangle[/spoiler] has the same energy as <span class="math">|0\rangle[/spoiler].
The Wigner representation of a symmetry is just that. It says that that indeed <span class="math">u(s)[/spoiler] is an (anti-)unitary operator and the energies are the same because, in fact, the <span class="math">u(s)|0\rangle=|0\rangle[/spoiler] because <span class="math">Q|0>=0[/spoiler] and so <span class="math">u(s)=1[/spoiler].
The Goldstone alternative now says: Yes, conserved current <span class="math">J[/spoiler] is there BUT... "<span class="math">Q|0>[/spoiler]" isn't even thing. Now you still have the Noether feature (the conserved current), but <span class="math">\int J^0 dV[/spoiler] is a shitty object. So here are now different states of the same energy but you don't transform along those. Happy birthday Goldstone bosons.
Now if you couple the above field (call it Higgs now) to the fermionic field and impose gauge invariance for the latter, you e.g. get the electromagnetic field with the relevant gauge symmetries of the Lagrangian, and (Higgs mechanism) the gauges you can have now happen to overwrite any Q-non-symmetry. The gauge does what a u(s)-transformation would do.
Now, if you go on and actually choose a gauge the you actually fix the Higgs ground state (that should be your answer), and the electromagnetic field is massive.

Related:
I'm working though a 6000 page collocation aiming at covering the easy parts of QFT, which moreover can actually be formulated fully rigorously. Always 700 pages math, 300 pages physics.
Just saw that the "classical electrodynamics" part is online:

http://pastebin.com/sZEXEcvc

>>6397768
this is immensely helpful. Thank you. Fucking saved.

>>6396308
what is the role of quantum measurement theory in quantum field theory?

>>6397768
You're the fucking man, OP.

>>6397774
Yeah, glad I could help. A more formal discussion of the "Q|0>"-part can be found here:

http://particle.univie.ac.at/studying-physics/

Summer semester 2010, around section 13 in the notes of Prof. Neufeld. funfact: The guy lectures with Schrödingers blackpoint pointer.<span class="math">{}^{citation\ needed}[/spoiler]

>>6397776
There is no point in excluding QFT when one says QM.

blackboard*

So as I understand it, gravity is successfully modeled as a space-time bending, but not quantised. Electromagnetism is quantised, but not modeled as a bending. What's with strong and weak forces then? Is it what ST tries to do? To both explain every force as a bending and quantise it? That string guy who talked with that populariser guy who has Feynman-like hair said that ST concurrents are nowhere near ST and that's why he's in ST. How true is this statement?

>>6397885
You mean the one Shrodinger used or is it a name for some extremely cool hi-tech thingy based on some Shrod. principle?

>>6397917
The former, it's a broken piece of wood.
funfact 2: Boltzmanns office is the PC room now.

>>6397768
I have part 1 of this series; is it worth working through if I don't know any QFT, but am familiar with QM and a good variety of modern mathematics.

>>6398089
It's extremely difficult for me to judge. Firstly, he does a lovely job of presenting the beauty of it, but he really really takes his time.* I knew the ideas before I read it and so I can't tell if you get the point if you don't invest time to get to the right parts.
I must say I'm very positively surprised by the book, but I don't quite understand how: The chapter are totally out of order and even sometimes repetitive. The merit is that it puts some concepts in a clear way I've never seen before (in particular, I love section 7 of the first book, the lattice treatment of phi^4-theory).
>...if I don't know any QFT
Yes, that's the point of an introduction to QFT ;) And that one is extremely easy on the eyes. If you want t a book that comes to the point, I hear people use this now
http://web.physics.ucsb.edu/~mark/qft.html
but I'm personally not extremely interested in doing quantum field theory.

*the second book "Quantum Electrodynamics", the quantization of the electromagnetic field (on discrete space approximation) starts - I shit you not - on page 881.

>>6397914
>Electromagnetism is quantized, but not modeled as a bending.
It's bending too, but not of velocity trajectories (not of spacetime itself). The electromagnetic field strength is the curvature of, roughly, the space of 'square root of the probability density' values. The other two forces are de facto the same as electrodynamics, just with more photons (then called W,Z-boson in the weak force case, of gluons in the strong force case).

>That string guy who talked with that popularize guy who has Feynman-like hair said that ST [rivals] are nowhere near ST and that's why he's in ST. How true is this statement?
If quantum gravity is New Zealand, then String theory is Australia, the other theories are European countries, but ships aren't invented yet.

What is this nerd shit?

>The other two forces are de facto the same as electrodynamics, just with more photons (then called W,Z-boson in the weak force case, of gluons in the strong force case).
But then why are they treated as separate forces? There must be some important fundamental differences or something.
>If quantum gravity is New Zealand, then String theory is Australia, the other theories are European countries, but ships aren't invented yet.
Can you use some other analogy, neither geography nor history are my strong suits?

>>6398165
which board are you from?

>>6398169
/lit/

>>6398179
it's advanced electronics

>>6397557
>If instead we consider a quantum field with that potential
In the sentence above, did you not do just the same thing?

>>6398153
>It's bending too, but not of velocity trajectories (not of spacetime itself). The electromagnetic field strength is the curvature of, roughly, the space of 'square root of the probability density' values. The other two forces are de facto the same as electrodynamics, just with more photons (then called W,Z-boson in the weak force case, of gluons in the strong force case).

I kind of disagree with this. First of all classical electrodynamics can be understand as a curvature on a fibre bundle afaik and I would not know how to introduce curvature on the space of 'square root of the probability density'. Furthermore, I definitively would not call these forces de facto the same as electrodynamics, for example the gauge group of electrodynamics is U(1) and that of chromodynamics SU(3), this results for example in the fact that gluons have self-interactions, but photons don't.

>>6398523
I meant exactly the <span class="math">C^n[/spoiler]-bundle where <span class="math">\psi[/spoiler] (the 'square root of the probability density') takes values in.
De facto the same means of course "the same but not exactly the same". The asker was suggesting the weak and the strong force aren't captured by standard QFT methods and that's the main reason string people have a job - I was contrasting that by saying they are really pretty similar.
From the sound of the question I didn't see a point is using the terms bundle and Yang-Mils type

>>6398523
I meant exactly the <span class="math">C^n[/spoiler]-bundle where <span class="math">\psi[/spoiler] (the 'square root of the probability density') takes values in.
De facto the same means of course "the same but not exactly the same". The questioner was suggesting the weak and the strong force aren't captured by standard QFT methods and that's the main reason string people have a job - I was contrasting his idea by saying that the used QFTs are actually very similar already.
From the sound of the question I didn't see a point in using the terms bundle and Yang-Mills type.

>>6398541
I thought the probabilty density interpretation fails in qft. The fiber bundle description works on the classical level. I'm not very solid in this fiber bundel business I should mention, to my own disdain.

>>6398541
>>6398555
pls respond

>>6398555
Might be you speak of the problems when computing one-particle relativistic wave functions.
If e.g. particle number is conserved and you consider a small amount of particles in a scattering process, and you get a non-vanishing cross section to many many states, then a frequentist interpretation for the transition amplitudes is sensible.
I put the square root in big quotes anyway - of course, it's not even a scalar object I speak of.

>The fiber bundle description works on the classical level.
Yeah, pretty sure that's how people think about it in any case. Good thing is that in path integral quantization, you don't have to pass to operator valued fields at all.

Super minor Language issue: Not sure if that's relevant saying, but
>...the fact that gluons have self-interactions, but photons don't.
catched my eye. Be aware that while there is no non-linear term in the EM Lagrangian there is still photon-photon interaction,
pic related

>>6398583
>Be aware that while there is no non-linear term in the EM Lagrangian there is still photon-photon interaction
Shit, you are right. Them higher than tree-level diagrams. Let me phrase it this way: there are no vertices in Feynman diagrams which have solely photons as incoming and outgoing particles, so to say there is no 'direct' photon interaction at one vertex (which is a 'shady' thing to say depending on how physical 'real' you see Feynman diagrams - which annoyed me pretty much in one lecture I had once)

>>6398583
>>6398601
Kannst du übrigens zufällig deutsch? Habe irgendwie den Eindruck bekommen aufgrund der links die du geposted hast.

>>6398607
Yes, but I'm going to sleep now.

PS: I'm not sure what exactly you mean is shady. But in any case, to me Feynman diagrams are literally just an algorithmic implementation of
http://en.wikipedia.org/wiki/Formal_power_series#Extracting_coefficients
and in any case, the Church-Turing thesis is about the only thing which I'd dare to consider "real".

>>6398583
Also can one give a rigorous definition of path integrals in cases where rick-rotation does not work?

>>6398629
Yes, to me it's also only this. But in my case the lecturer always said stuff like, "and then we have a electron traveling here, sending out a photon and absorbing it later blabla..".
Oh well, sleep well.

>>6398629
Btw it was nice talking to you.

>>6398629
The "physical" Church-Turing thesis?

>>6398496
no? A quantum particle (in this case, 2 degrees of freedom) is not a quantum field (infinite degrees of freedom)

>>6398541
> the weak and the strong force aren't captured by standard QFT methods
Monte-Carlo and lattice simulations aren't standard QFT methods?

>>6399126
>The questioner was suggesting the weak and the strong force aren't captured by standard QFT methods

>>6398640
>can one give a rigorous definition of path integrals in cases where rick-rotation does not work?
The stack exchange thread "Path integral vs. measure on infinite dimensional space" might provide you with some answers.
Indeed, this question
>>6395317
is motivated by this answer
http://physics.stackexchange.com/questions/1894/path-integral-vs-measure-on-infinite-dimensional-space/13315#13315

>>6398644
Regarding "shady": I found your rewording of "nonlinear" a little unnecessary.
>Shit, you are right. Them higher than tree-level diagrams. Let me phrase it this way: there are no vertices in Feynman diagrams which have solely photons as incoming and outgoing particles, so to say there is no 'direct' photon interaction at one vertex (which is a 'shady' thing to say depending on how physical 'real' you see Feynman diagrams - which annoyed me pretty much in one lecture I had once)
You know how you can read off the vertex structure from the Lagrangian right? Power counting of fields at the interaction term (marked by the coupling <span class="math">\lambda[/spoiler]) gives you the order and shape (colour) of the internal vertices. Having <span class="math">\propto {\lambda_\kappa} \phi^4[/spoiler] means <span class="math">\phi^4[/spoiler]-theory vertices are connected by four lines. Having <span class="math">\propto {\lamdba_e} {A_\mu} {J^\mu}[/spoiler] with <span class="math">{J_\mu} = \psi{\gamma_\mu}\phi[/spoiler] means QED Feynman diagrams consist of dots with one wiggly photon lines for <span class="math">A[/spoiler] and two fermion lines for <span class="math">\psi[/spoiler].
So instead of that description above, you could have just said "there is no <span class="math">A^{3+m}[/spoiler]-term in the Lagrangian with <span class="math">m\ge 0[/spoiler]"
i.e., there is no non-linear term.

>>6398676
not certain what you mean, but I think the answer is yes.

>>6398640
>can one give a rigorous definition of path integrals in cases where rick-rotation does not work?
The stack exchange thread "Path integral vs. measure on infinite dimensional space" might provide you with some answers.
Indeed, this question
>>6395317
is motivated by this answer
http://physics.stackexchange.com/questions/1894/path-integral-vs-measure-on-infinite-dimensional-space/13315#13315

>>6398644
Regarding "shady": I found your rewording of "nonlinear" a little unnecessary.
>Shit, you are right. Them higher than tree-level diagrams. Let me phrase it this way: there are no vertices in Feynman diagrams which have solely photons as incoming and outgoing particles, so to say there is no 'direct' photon interaction at one vertex (which is a 'shady' thing to say depending on how physical 'real' you see Feynman diagrams - which annoyed me pretty much in one lecture I had once)
You know how you can read off the vertex structure from the Lagrangian right? Power counting of fields at the interaction term (marked by the coupling <span class="math">\lambda_q[/spoiler]) gives you the order and shape (colour) of the internal vertices:
Having <span class="math">\propto {\lambda_\kappa} \phi^4[/spoiler] means <span class="math">\phi^4[/spoiler]-theory vertices are connected by four lines.
Having <span class="math">\propto {\lambda_e} {A_\mu} {J^\mu}[/spoiler] with <span class="math">{J_\mu} = \psi{\gamma_\mu}\phi[/spoiler] means QED Feynman diagrams consist of dots with one wiggly photon lines for <span class="math">A[/spoiler] and two fermion lines for <span class="math">\psi[/spoiler].
So instead of that description above, you could have just said "there is no <span class="math">A^{3+m}[/spoiler]-term in the Lagrangian with <span class="math">m\ge 0[/spoiler]"
i.e., there is no non-linear term.
(Going for a run.)

>>6398676
not certain what you mean, but I think the answer is yes.

>>6398640
>can one give a rigorous definition of path integrals in cases where rick-rotation does not work?
The stack exchange thread "Path integral vs. measure on infinite dimensional space" might provide you with some answers.
Indeed, this question
>>6395317
is motivated by this answer
http://physics.stackexchange.com/questions/1894/path-integral-vs-measure-on-infinite-dimensional-space/13315#13315

>>6398644
Regarding "shady": I found your rewording of "nonlinear" a little unnecessary.
>Shit, you are right. Them higher than tree-level diagrams. Let me phrase it this way: there are no vertices in Feynman diagrams which have solely photons as incoming and outgoing particles, so to say there is no 'direct' photon interaction at one vertex (which is a 'shady' thing to say depending on how physical 'real' you see Feynman diagrams - which annoyed me pretty much in one lecture I had once)
You know how you can read off the vertex structure from the Lagrangian right? Power counting of fields at the interaction term (marked by the coupling <span class="math">\lambda_q[/spoiler]) gives you the order and shape (colour) of the internal vertices:
- Having <span class="math">\propto {\lambda_\kappa} \phi^4[/spoiler] means <span class="math">\phi^4[/spoiler]-theory vertices are connected by four lines.
- Having <span class="math">\propto {\lambda_e} {A_\mu} {J^\mu}[/spoiler] with <span class="math">{J_\mu} = \psi{\gamma_\mu}\psi[/spoiler] means QED Feynman diagrams consist of dots with one wiggly photon lines for <span class="math">A[/spoiler] and two fermion lines for <span class="math">\psi[/spoiler].
So instead of that description above, you could have just said "there is no <span class="math">A^{3+m}[/spoiler]-term in the Lagrangian with <span class="math">m\ge 0[/spoiler]"
i.e., there is no non-linear term.
(Going for a run.)

>>6398676
not certain what you mean, but I think the answer is yes.

What happened to Josef and physics guy?

I don't know anything about QFT, but this is the kind of threads /sci/ needs/ Good job OP.

>>6399255
He's back to germany and very active in the Haskell community.

>>6399255
Josef is back to Germany and very active in the Haskell community.

>>6399232
The rewording was to explain in a more physical sense why I originally said there is no photon-photon interaction (though one could argue about the physical relevance of Feynman diagrams, I know)

>You know how you can read off the vertex structure from the Lagrangian right?
Yes, but why do you say A^(3+m) with m >= 0, if m=-1 the term is still non-linear (or is it because you had renormalization in mind - talking about stuff I have no idea about but effective theories are not necessarily renormalizable, right?)

>>6399232
>The stack exchange thread "Path integral vs. measure on infinite dimensional space" might provide you with some answers
Okay but the problem is (besides the fact that I suck in measure theory), that this person is still talking about euclidean continuation, but afaik there are certain situations where you can't do that.

>>6399687
>active in the Haskell community
I always did say he is destined for mediocrity.

>>6400068
A vector to the power of -1?
But yeah, it's only linear (in the field equations) if the exponent is 1 or 2.
Btw. if you want a theory which is silly in terms of non-linearity, check out
http://en.wikipedia.org/wiki/Sine_gordon_equation

pic related: reality fucks with my brain

Regarding renormalization and friends, I watch
http://ocw.mit.edu/courses/physics/8-851-effective-field-theory-spring-2013/video-lectures/lecture-1-introduction-to-effective-field-theory-eft/
tonight and might be continuing it the next week. Feel free to join.

>>6400080
I didn't intend to point you to that answer in particular, but the thread in general. I couldn't tell you more than they say there. And what they say is that it works only for certain unphysical theories, e.g. 2dim models.
And it's your job to make it right ;)

>>6400113
Never mind the m=-1 case, I got confused.

>pic related: reality fucks with my brain
That picture is really crazy.

>http://ocw.mit.edu/courses/physics/8-851-effective-field-theory-spring-2013/video-lectures/lecture-1-introduction-to-effective-field-theory-eft/
That's certainly nice, but my internet sucks pretty bad atm. Also I am afraid I'm not ready for it now.

>And it's your job to make it right ;)
Tehe, probably more realistic for you to do that.

>>6400142
where do you live?

>>6401299
pic related (yours too, I guess)

What is quantum field theory and why do we need it?

>>6401650
To quantise fields via apparatus of the modern physics. To, using mathematical theory, create order out of chaos. If that's not hot, I don't know what is.

>>6401676
What does it mean to "quantise" a field?

>>6401677
To do the quantum science maths theory thingy to the field.

>>6396369
>it's difficult to quantify

MFW Dat pun.

>>6401699
What's that?

>>6401677
"quantizing" the field basically means turning the field into an operator-valued field, and then imposing commutation relations based on the poisson brackets.
This is the same thing that is done in non-relativistic QM. <span class="math">x_{i}[/spoiler] and <span class="math">p^{i}[/spoiler] are turned into operators <span class="math">\hat{x_{i}}[/spoiler] and <span class="math">\hat{p^{i}}[/spoiler], with the commutation relations: <span class="math"> \left [ \hat{x_{i}}, \hat{p^{j}} \right ] = \delta_{i}^{j} [/spoiler]

>>6401738
whoops, the commutation relations should be:
<span class="math"> \left [ \hat{x_{i}}, \hat{p^{j}} \right ] = i \hbar \delta_{i}^{j} [/spoiler]

How does this whole running coupling constant work? and what's renormalization group flow? I only vaguely remember how renormalization itself works, something with introducing a cutoff / switching the number of dimensions to n+ epsilon, then absorbing the terms that would blow up depending on the reularator into coupling constants.

>>6401650
>>6401676
From the preface to Volume 1 of Weinberg's "Quantum Theory of Fields":
>Why should we believe in the canonical quantization or path integration? Why should we adopt the simple field equations and Lagrangians that are found in the literature? For that matter, why have fields at all? It does not seem satisfactory to me to appeal to experience; after all, our purpose in theoretical physics is not just to describe the world as we find it, but to explain - in terms of a few fundamental principles - why the world is the way it is.
>The point of view of this book is that quantum field theory is the way it is because (aside from theories like string theory that have an infinite number of particle types) it is the only way to reconcile the principles of quantum mechanics (including the cluster decomposition property) with those of special relativity.
(I'm not OP)

>>6401805
> Quantum field theory arose out of our need to describe the ephemereal nature of life.

- QFT in a Nutshell, Zee