Jeff Harvey
|
Registered User
|
Professor of Physics at the University of Chicago
|
|
20h |
comment |
Derivation of Bessel functions Not a comment on the math, but if you really want to reformulate Quantum Mechanics in terms of classical fluid dynamics and want to be taken seriously rather than viewed as a crank then you are first obligated to understand how Quantum Mechanics is currently formulated and used in some detail. |
|
May 14 |
comment |
What is the “fundamental theorem of invariant theory” ? @user6818 I think you need to read the paper you are citing more carefully. They do specify the $SU(N_c)$ representations on the bottom of p. 9 and top of p.10. They consider four cases and in each case they specify the $SU(N_c)$ representation content (the $R_i$ in their notation). |
|
May 14 |
comment |
What is the “fundamental theorem of invariant theory” ? @user6818 As in the question is not well posed to start with and second it is not written in language that most mathematicians will understand. I partially understand what you are asking because I happen to be a physicist. I'd suggest that you either ask the question on physics stack exchange or make the effort to translate your question into a precise mathematical question framed in language that mathematicians will understand. Otherwise your question will be and should be closed since this is a site for research level math questions. |
|
May 13 |
comment |
What is the “fundamental theorem of invariant theory” ? Your question doesn't contain enough information for a sensible answer until you also specify the $SU(N_c)$ representation of the fields and also their statistics (bosons or fermions). |
|
May 1 |
comment |
Is there a “right” proof of Riemann’s Theta Relation? It is so annoying not to be able to edit comments! Please in the above read $t A A$ to be the transpose of $A$ times $A$ and interpret the $\frac{1}{2}$ as a prefactor in front of a $4 \times 4$ matrix. |
|
May 1 |
comment |
Is there a “right” proof of Riemann’s Theta Relation? Mumford explains that the relation depends on a matrix $A$ satisfying $tAA=I$ with $I$ the identity matrix ( I have chosen $m=2$ in his notation on p.14 of Tata I). I think you must have a typo because your matrix does not obey this identity. I believe you want $$ A= \frac{1}{2} \begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 \end{matrix} $$ What is special about this matrix is that it acts as a triality transformation on the weight space of so(8)$, transforming vector weights into spinor weights. This is related to my answer to this question. |
|
Apr 13 |
comment |
Is there a “right” proof of Riemann’s Theta Relation? Isn't the usual $\theta$ function as defined in Mumford's Tata lectures in the first chapter $\theta: {\mathbb C} \times {\mathbb H} \rightarrow {\mathbb C}$ with ${\mathbb H}$ the upper half plane? What is your "usual" theta function? |
|
Apr 13 |
answered | Is there a “right” proof of Riemann’s Theta Relation? |
|
Apr 13 |
comment |
Is there a “right” proof of Riemann’s Theta Relation? There is a nice interpretation in terms of characters of affine $Spin(8)$ that involves triality and is related to supersymmetry in string theory. I can write up some details tomorrow unless someone else beats me to it. |
|
Apr 10 |
comment |
Why are currents named currents? To a physicist it is strange to see a current in a Lorentz invariant theory written as a 2-form in space rather than as a 3-form in space-time. |
|
Mar 27 |
comment |
Algebraic independence of $E_2$, $E_4$ and $E_6$ Dear Jonas, To expand slightly on Emerton's comment, $E_4$ and $E_6$ are modular forms, so in particular $E_4(-1/\tau)=\tau^4 E_4(\tau)$ and $E_6(-1/\tau)= \tau^6 E_6(\tau)$. On the other hand $E_2$ is not a modular form, it is only quasimodular. It obeys $E_2(-1/\tau)= \tau^2 E_2(\tau)- 6 i \tau/\pi$. No algebraic combination of $E_4,E_6$ can transform this way under $\tau \rightarrow -1/\tau$ so $E_2$ is algebraically independent of $E_4,E_6$. |
|
Mar 13 |
comment |
Geometric treatment of the Ward-Takahashi identity In many cases no invariant measure exists. There are anomalies. The passage from finite dimensional integrals to path integrals is more subtle than you indicate here. |
|
Mar 13 |
comment |
Geometric treatment of the Ward-Takahashi identity Your questions tend to be too terse in my opinion. Some context would be helpful. Why in the context of differential geometry? Modern from what point of view? Many symmetries of classical field theory Lagrangians do not survive in the quantum theory. There are anomalies, aka anomalous Ward-Takahashi identities. This is actually a huge subject so it is hard to answer without knowing in more detail what you are actually interested in. |
|
Mar 11 |
comment |
Applications of n-dimensional crystallographic groups I added a few early references. |
|
Mar 11 |
revised |
Applications of n-dimensional crystallographic groups added references |
|
Mar 11 |
answered | Applications of n-dimensional crystallographic groups |
|
Mar 4 |
comment |
What’s about “quantum modular forms”? Zagier's paper on quantum modular forms is available as research article #120 on his website: people.mpim-bonn.mpg.de/zagier |
|
Feb 22 |
comment |
How should a professor feel peace of mind when a student leaves academia? Do you have children? |
|
Jan 31 |
comment |
Mock modular forms and (indefinite) quadratic forms I don't know the answer to your question, but one place that Appell-Lerch sums show up is in the Polar part of meromorphic Jacobi forms. The recent paper arXiv:1208.4074 by Dabholkar, Murthy and Zagier has a detailed treatment of the relation between meromorphic Jacobi forms and mock modular forms along with many examples. |
|
Jan 30 |
comment |
Mock modular forms and (indefinite) quadratic forms Why isn't there any $z$ dependence on the rhs of your equation for $f(q,z,-1)$? Also, can you provide details on the relation between f(q,z,1) and a specific mock modular form? |
|
Jan 28 |
comment |
Inclusion of information about external particles to calculate scattering amplitudes in string theory I have my doubts that this is a good question for MO as it is standard textbook material. String backgrounds determine a CFT, in this CFT there is a state-operator correspondence and the vertex operators used in string scattering computations are given by this correspondence in terms of the external scattering states one is interested in. This is discussed in Chapters 2,3 of volume I of Polchinski's book on string theory. Perhaps you want something else though as I find the mention of bound states and lifetimes and the integral over $d\tau$ confusing. Where did you get your schematic equation? |
|
Jan 27 |
comment |
Quantization of a classical system (e.g. the case of a billard) I see no reason at all why one needs to have a classical Hamiltonian as a starting point for the description of an intrinsically quantum system. The logical map is quantum --> classical in physics, not the other way around since the world is fundamentally described by quantum mechanics and classical behavior only arises in a limit. Classical systems do play a more fundamental role in defining measurement in quantum mechanics, but that is a different and more complicated can of worms. |
|
Jan 27 |
comment |
Quantization of a classical system (e.g. the case of a billard) @Uwe If you are a physicist then the "right" quantization is determined by comparison to experiment. |
|
Jan 17 |
accepted | Relation between TQFT and Wilson lines, boundary conditions, surface defects etc |
|
Jan 16 |
answered | Relation between TQFT and Wilson lines, boundary conditions, surface defects etc |
|
Jan 8 |
revised |
Meaning of a phrase from “The algebra of grand unified theories”. additional explanation added |
|
Jan 8 |
answered | Meaning of a phrase from “The algebra of grand unified theories”. |
|
Dec 28 |
comment |
What do correlation functions compute in CFT? Of course CFT also shows up in string theory where the fields have a different interpretation, but I thought I would give you the historical origins of CFT since it has the most direct physical interpretation. |
|
Dec 28 |
answered | What do correlation functions compute in CFT? |
|
Dec 28 |
comment |
What do correlation functions compute in CFT? Or are you asking about a purely mathematical interpretation of the correlation functions, which is how I read your question? |
|
Dec 24 |
comment |
Beginner question on constraints of a wave function in quantum mechanics I don't think you are describing Griffith's argument correctly. He does not claim that $\psi \rightarrow 0$ as $x \rightarrow \infty$ as a consequence of the time independence of the normalization of $\psi$. He claims $\psi \rightarrow 0$ as a consequence of $\psi$ being normalizable. However he then has a horrible footnote warning about good mathematicians with pathological counterexamples and saying that in physics the wave function always goes to zero at infinity. Except of course he later uses a formalism for scattering problems where $\psi(x,t)$ behaves as $e^{ikx}$ at infinity. ;) |
|
Dec 18 |
answered | Explanation for E_8’s torsion |
|
Dec 17 |
comment |
The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch? @Alexander Chervov The Dyson quote is from his "Missed Opportunities" Gibbs lecture in 1972. He says "As a working physicist, I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce. Discussing this divorce, the physicist Res Jost remarked the other day, `as usual in such affairs, one of the two parties has clearly got the worst of it.'" Dyson clearly thought that physics had got the worst of the divorce. |
|
Dec 14 |
awarded | ● Nice Answer |
|
Dec 14 |
comment |
Mathematician trying to learn string theory It would be fun, but would require a lot of time and dedication from both parties. I'm tempted to ask the converse, how does a physicist who knows QM, QFT and string theory learn algebraic geometry, or at least the parts that are most relevant to string theory? The standard answer seems to be to read the first few chapters of Griffiths&Harris and lecture notes by Candelas and others but I wonder if there is a better answer that doesn't involve a willing algebraic geometer. |
|
Dec 13 |
answered | Mathematician trying to learn string theory |
|
Dec 13 |
comment |
Mathematician trying to learn string theory Understanding Lagrangians and symmetries is important, but some theories don't have Lagrangians. In fact such theories, like the (2,0) theory in six dimensions, are the focus of much recent research. Also, while Petr Horava has done much excellent work, including work that foreshadowed the discovery of D-branes, Joe Polchinski is the person who is generally credited with discovering D-branes in string theory. |
|
Dec 9 |
awarded | ● Guru |
|
Dec 9 |
accepted | Does Physics need non-analytic smooth functions? |
|
Dec 4 |
comment |
Is there “harmonic potential” for classical bosonic string? There seem to be many gins with <10 reputation points. It is not necessary to create a new id every time you ask a question. |
|
Dec 3 |
comment |
Stringy version of RR Barton Zwiebach's book "A first course on string theory" will give you the simplest answer to this question and your other question. This question is not really appropriate for MO. Also, you should not answer your own question with a follow-up question. |
|
Nov 28 |
comment |
Does Physics need non-analytic smooth functions? Yes. My understanding of the phrase "one part in $10^x$" is that it is equivalent to saying an accuracy of $10^{-x}$. For a detailed statement of the comparison between theory and experiment for g-2 see arxiv.org/pdf/1205.5368.pdf |
|
Nov 28 |
revised |
Does Physics need non-analytic smooth functions? improved grammar |
|
Nov 26 |
awarded | ● Good Answer |
|
Nov 26 |
comment |
Quantum mechanics basics When you quantize the electromagnetic field the wave function is most definitely not the electric and magnetic fields. The fields are operators which act on a complex valued wave function, just as in the non-relativistic Schrodinger equation. You are confusing complex valued fields with the complex valued wave function of quantum mechanics. They are two different things. |
|
Nov 26 |
awarded | ● Mortarboard |
|
Nov 26 |
awarded | ● Nice Answer |
|
Nov 26 |
answered | Does Physics need non-analytic smooth functions? |
