Ok, so the correct interpretation is that what was thought of as the one-particle operator on the state space (schrodinger picture), is a actually the field equation for the Heisenberg field operator
Since the many body Heisenberg Hamiltonian operator is diagonal in this basis, if I transform to the Schrodinger picture and isolate to one particle (a normal particle or antiparticle which i can because the Hamiltonian is diagonal), I should get $H^{(1)} \psi = E \psi$ with $E$ always being positive where $H^{(1)}$ is the one body Hamiltonian. Is this correct?
@Bml the introductory bits, i.e. very much pre-uni, cannot be done to any uni-level sense of goodness. The teacher is trying very hard to fight against beginner ignorance, and if you try to do something sophisticated, the student will be completely lost
@Allie Prior to the EL equations step, you have yet to specify any path at all. The derivation of the EL equations itself you have a the Lagrangian's dependencies left unspecified (so that the position and velocities can be varied independently at this step) and the statement of the EL equations assert a condition that stationary paths have to satisfy. Solving the EL equations thus is about finding the stationary path(s). No mention of evolution yet.
@Madder It might be easier to directly obtain Equation (6.2.15) $$\begin{align}1&=\sinh2K^*\sinh2L\\&=2\sinh K^*\cosh K^*\sinh2L\\&=\tanh K^*\cosh^2K^*2\sinh2L\\\frac1{\cosh^2K^*}&=e^{-2L}(e^{+2L}-e^{-2L})\\1-\tanh^2K^*&=1-e^{-4L}\end {align}$$ which is manifestly true, so that this symmetry must be how the duality is motivated.
@Allie Perhaps what you're fishing for this is: The extremization of the action is a boundary value problem where you are seeking the stationary path of the action for any fixed starting and ending point. It's not an evolution procedure that would start from initial conditions. But the E-L equations are local differential equations, which now can be solved as an initial value problem for specified initial condition $q(t_0),\dot{q}(t_0)$
I read last night that there is a duality between the 1D quantum transverse ising model and the 2D classical ising model. Is this true? I would think solving 1D quantum ising is much easier than 2D classical (no external field).
Many other places don't really seem to teach "classical mechanics" as we would understand it (i.e. analytical mechanics with Lagrangians and Hamiltonians) at all in undergrad. That's also the reason you see so many weird "classical field theory" intros in QFT books because they're written for people who have never seen a classical field theory before (!!!)
Physics has become so broad that a lot of the fundamental topics are not much use to many students compared to very specific applied topics that they will use everyday
Like compuational techniques I imagine are used by everyone
Suppose you wish to find the path, say, from a ball at the bottom of a hill to the top. Obviously this wont occur naturally. In that case will there be no stationary path? Or something else
@ACuriousMind My analytical mechanics course was not very rigorous and I still have some doubts about the way my prof explained some things about HJ equation but it was so cool to end the course with some field theory and then apply it to the non-linear Maxwell equations
For those interested, by non-linear Maxwell equation, I mean the construction of the lagrangian that corresponds to the effective interaction arising from box diagrams
@ACuriousMind just it's also extremely surprising to me also that class. mech would be optional or that it wouldn't be done before QFT anywhere! didn't you say the other day "if you haven't seen symmetries discussed before a QFT course..."?
@ACuriousMind we did two courses on class. mech, an intro one that was langragians mostly and one that did more Hamiltonian mech and chaos theory. symmetries were covered but not in great detail - no Noether
@DIRAC1930 I don't know how they did but the idea is to understand which scalars are availabe to construct your lagrangian and include the relevant ones. The Maxwell term is the trace of $F^2$, then the next on the list are like trace of $F^4$ (also consider the dual tensor)
Another way is to build an effective lagrangian with QED, you can find that on Schwartz
Left is what I will call a simple closed graph representing a nonzero contribution to the 2D ising model partition function.
On the right is the Kramers-Wannier dual graph (color coded: purple is the induced dual lattice and pink is the induced dual graph).
Feynman writes if there are $L$ unlike bonds (an edge in the dual graph between a +1 and -1), then there are $2N - L$ like bonds where $N$ is the number of original lattice points (I think).
By order you mean Perturbation order i.e 1st 2nd etc? Additionally, you say: " By putting in a spurious polarization and checking your amplitude is zero for it, you show that's true. " What is true? gauge invariance or that we are not missing any additional feynman diagram, or both?
"Ok, so the correct interpretation is that what was thought of as the one-particle operator on the state space (schrodinger picture), is a actually the field equation for the Heisenberg field operator Since the many body Heisenberg Hamiltonian operator is diagonal in this basis, if I transform to the Schrodinger picture and isolate to one particle (a normal particle or antiparticle which i can because the Hamiltonian is diagonal), I should get $H^{(1)} \psi = E \psi$ with $E$ always being positive where $H^{(1)}$ is the one body Hamiltonian. Is this correct? $\psi$ is a one particle state h…
My question is that $\hat{H}(\hat{\phi}(\mathbf{x},t))$ is the Hamiltonian in the Heisenberg picture. In the Schrodinger picture, one will get an operator $\hat{H}(\hat{\phi}(\mathbf{x}))$ acting on the state space $| \psi(t) \rangle$. This Hamiltonian is diagonal so I can essentially consider a one particle subspace. If I do this, will I get $H^{(1)} \psi(t) = E \psi(t)$ with $E$ being always positive
@ACuriousMind It doesn't seem to me very intuitive or "at your face" that we consider uniform distribution. Is that an arbitrary choice or there's an argument for that?
I wouldn't choose anything or propose anything, since I have no clue what is happening. Whether in books or in notes I take, things are just in the air, just some QED process w/e that may be and somehow incoming electrons and photons average over their spin and polarization, and that is the average of continuous uniform distribution
I mean how on earth, can I even think of this
the electrons have $\pm \frac 1 2$ spin values
and it's also discrete
variable
so continues uniform distribution wouldn't make sense here
I don't know or understand the exact context of your question. I just want to point out that it is more or less natural to choose the uniform distribution if you don't have any knowledge apart from the theoretically possible events that might occur
If I give you a dice, and you should guess a probability distribution, i.e. you should tell me the probability for the $n$-th side to occur, what would you tell me?
Why do certain random strings produce colors when entered as background colors in HTML?
For example, bgcolor="chucknorris" produces a red background:
<body bgcolor="chucknorris"> test </body>
Conversely, bgcolor="chucknorr" produces a yellow background:
<body bgcolor="chucknorr"> t...
It seems that the whole point of obtaining quantised fields, is so that you can get a quantised Hamiltonian $H(\hat{\phi},\hat{\pi})$ that acts on the state space exactly analogous to QM
And that it also allows one to motivate certain observables $\mathcal{O}(\hat{\phi}, \hat{\pi}$) up to operator ordering ambiguities
maybe I'm just hoping for someone too convenient, but it'd be cool if someone develops a way to compare a given paper with the rest of the literature and find which other papers are the most related
maybe technology will reach that point in a decade or two
@HerrFeinmann well I'm hoping papers of the future take full advantage of digitization. E.g. plots you can interact with, so the reader can glean more info than what a PDF could show with a mere image
@SirCumference I think papers which want to do that sort of thing link to something like a GitHub repo or something hosted by the publisher. it's not common but I think I've heard of it. pdf is just more appropriate as the main format I think
@qwerty well, imagine if papers had clickable checkboxes that let you hide or show certain elements of plots, or sliders for the reader to change the axes limits however they want
has anyone here heard of the internet puzzle/game called whentaken? it's like geoguessr but from a static photo and you also have to guess the year it was taken
Why do certain random strings produce colors when entered as background colors in HTML?
For example, bgcolor="chucknorris" produces a red background:
<body bgcolor="chucknorris"> test </body>
Conversely, bgcolor="chucknorr" produces a yellow background:
<body bgcolor="chucknorr"> t...
English Language & Usage: Multi-Layer…
English Language & Usage: Multi-Layer…
