# Tag Info

19

These are all good questions. Perhaps I can answer a few of them at once. The equation describing the violation of current conservation is $$\partial^\mu j_\mu=f(g)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$ where $f(g)$ is some function of the coupling constant $g$. It is not possible to write any other candidate answer by dimensional analysis ...

14

The easiest way to see imaginary time used is in elementary quantum mechanics in one dimension. (This is the explanation cribbed from wikipedia). Suppose we're looking at a tunneling-through-a-barrier problem. We start with the Schrodinger equation: $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x) = E\psi(x)$$ Make the ansatz $$\psi(x) ... 9 I will add to twistor59 answer. Hawking liked the concept of imaginary time \tau=\mathrm{i}t because it transforms a Lorentzian metric$$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$into a four dimensional like Euclidean metric$$ds^2 = +c^2 d\tau^2 + dx^2 + dy^2 + dz^2$$Hawking and others believed that a quantum gravity theory could be developed in this ... 9 Let us look at the instantons of an ordinary pure Yang-Mills theory for gauge group G in four Euclidean dimensions: An instanton is a local minimum of the action$$ S_{YM}[A] = \int \mathrm{tr}(F \wedge \star F)$$which is, on \mathbb{R}^4, precisely given by the (anti-)self-dual solutions F = \pm \star F. For (anti-)self-dual solutions, ... 5 It's Stokes's theorem. Consider a field F = dA + A \wedge A such that A is pure gauge at infinity, that is, \lim_{x\to\infty} A(x) = \omega\, d \omega^{-1} for some \omega : S^3 \to SU(2) \sim S^3 where \omega is a function on the 3-sphere because the limit can depend on the direction out to infinity. In differential forms the first expression is ... 5 The reason why classical solutions add a "lot" to the path integral is that their action (phase) is stationary i.e. almost the same phase as the action (phase) in their reasonably large vicinity of the configuration space; one gets positive interference as a consequence. More generic paths cancel with the adjacent ones whose phases are different and random. ... 4 I) This is discussed around eq. (23.7.1) on p. 462 in Ref. 1. The task is to perform the path integral$$\tag{1} \int_{BC} [d\phi]e^{\frac{i}{\hbar}S[\phi]} ~=~\sum_{\nu}\int\! du \int_{BC_0} [d\phi_q]e^{\frac{i}{\hbar}S[\phi_{cl}+\phi_{\nu,u}+\phi_q]} $$over fields \phi with some (possible inhomogeneous) boundary conditions BC. This is done by ... 4 This is apparently too long for a comment, so it's going to be fleshed out into at least (hopefully) a partial answer. A big problem I see with this is determining which theories can live in the same universe. In this way, I think Liang Kong's Mathematical Theory of Anyon Condensation provides a way forward. To reconstruct your set up: Consider three ... 4 Fermionic zero modes on an instanton background are in one-to-one correspondence with solutions to the Dirac equation$$(i D_\mu \gamma^\mu - m)\Psi = 0$$where the partial derivative D_\mu contains the gauge field term with the gauge field defining the instanton solution substituted into it. We want to study these solutions in the Euclidean spacetime. ... 4 For the pure e.g. d=4 gauge theory instanton and gauge field perturbations around it, there is no negative mode – the counterpart of the bound state. It's the only one among the 3 classes that is absent here. One may see this absence by noticing that the gauge theory may be embedded into a supersymmetric theory with the same gauge-field degrees of ... 4 A soliton is a localized, non-dispersive solution of a nonlinear theory in Euclidean space. It certainly is a real object: you have a famous story about a certain John Russell who observed soliton-like waves made by a boat on a river (wikipedia knows everything about it!) The so-called morning glory clouds in Australia ... 3 There is no better definition than what Wikipedia offers - in general, a topological excitation is a (field) state, i.e. a localized quantity since fields depend on spacetime, whose integral is a topological invariant. One prime example are Yang-Mills theories in 4D, where the integral \int \mathrm{Tr}(F\wedge F), as essentially the second Chern class of ... 3 The sphaleron is kind of the opposite of the instanton, and kind of the same. Let's make that statement precise: An instanton is a local minimum of the action that mediates vacuum tunneling (link to an answer of mine how and why instantons do that). The sphaleron sits in-between the vacua, in a certain sense, it is the instanton "in the middle of ... 3 Instanton calculations involve integrals over collective coordinates. One of these is the instanton size \rho. Reliable instanton calculations are those for which the integral over instanton sizes is dominated by small sizes so that (for asymptotically free theories) the coupling constant is small and higher order corrections in the semi-classical ... 3 Yes, you can recover the Seiberg-Witten prepotential from the Darboux coordinates X_\gamma (and indeed also from their "semiflat" versions X_\gamma^{sf}). The reason is the asymptotic property X_\gamma \sim exp(\pi R Z_\gamma / \zeta) as \zeta \rightarrow 0 (up to a \zeta-independent constant). Thus knowing X_\gamma is sufficient to recover ... 2 TO have an instanton solution, you need to map the (euclideanized) "spacetime at infinity" to the group manifold. In the case of SU(2), both the spacetime at infinity and the group manifold are S^3 and instantons are characterized by the integers. I hope you understand that much, at least for SU(2). If you're interested in 4d instantons, they are ... 2 I think the path-integral is a complete red herring here! I'll try to convince you that Wick rotation yields completely equivalent way of writing the Lagrangian in classical field theory. Consider a classical action$$S[x] = \int L[x(t)] dt where $x:\mathbb{R} \to \mathcal{M}$ for some target manifold $\mathcal{M}$. The Lagrangian is schematically given ...

2

Where a string carves out a $2$-dimensional world-sheet and a point particle carves out a $1$-dimensional world-line of spacetime, the instanton carves out a $0$-dimensional world-point. Counting only spatial dimensions, a string is $1$-dimensional and a point particle is $0$-dimensional. By logical extension, an instanton has dimension $-1$, if we only ...

2

Okay, I cannot give you a full understanding of what is going on, but I can make the objects we are dealing with more precise: There are two spaces here: The moduli space $M_\text{sh}(r,k)$ of framed torsion-free coherent sheaves of rank $r$ and second Chern class $k$ on the projective scheme $\mathbb{P}^2$ viewed as a complex analytic space with its ...

2

1) If $\phi$ is non-singular, you can safely multiply both sides by $\phi$ and get $\square \phi = \phi*0 = 0$. If $\phi$ is singular you can't do this because $\phi * 0$ is undefined. Equivalently, where $1/\phi=0$ you can have $\square\phi$ nonzero, as shown in (4.64). The singularities in $\phi$ will map to zeroes in $1/\phi$. 2) I believe only the ...

2

I'm not sure why you are asking, because you seem to mention the answer already. This problem has been studied thoroughly in the late 70's by Belavin et al and 't Hooft. As far as I understand, the quantum vacuum is the lowest energy eigenstate of a Hamiltonian. It turns out that the classical solutions to the equations of motion (of a particle, or a field) ...

1

1) NO 2) The usual way of doing it is to first solve the instanton solution in euclidean time, which is equivalent to obtaining soliton solution of a given potential. Since you have read the book, I am not going to explain how it is done for this case. Then, plugin your instanton solution to the euclidean action and evaluate it. Since ...

1

Let me first refer you to three references pedagogically treating Instantons in quantum mechanics: 1)Riccardo Rattazzi's lecture notes treating instantons in nonsupersymmetric quantum mechanics. In these notes the anharmonic oscillator model is elaborated with great detail 2) Philip Argyres lecture notes treating instantons in supersymmetric quantum ...

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There seems to be a slight confusion about the meaning of solution: The principle of least action leads to the equation of motion (Euler-Lagrange equation), which correspond to a minimum of the action functional. These equations can have multiple solutions, so there is no contradiction in the formalism. There can multiple solutions that minimize the energy, ...

1

A vacuum is a field configuration that is pure gauge, i.e. $A = g^{-1}\mathrm{d}g$ for a gauge transformation $g$, and hence $F = \mathrm{d}_A A = 0$ (for $\mathrm{d}_A$ the gauge covariant derivative). An instanton is a local minimum of the action, which is given by an (anti-)self-dual configuration $F = \pm \star F$. It is not a vacuum for non-zero ...

1

I know this is a year old question, but I am going to attempt an answer. As far as I can tell, this is not really a caveat. The reason for this is that I can always set the overall phase of the quark mass determinant to be zero with a chiral U(1) transformation. For a discussion of this see for example the chapter on theta vacua in Weinberg's QFT book. The ...

1

Maybe this would be better as a comment, since it is not a full answer, but I don't have enough reputation for that. The most important ambiguity is that there is an infinite number of functions that have the same asymptotic expansion. As an example, if $f(g)$ has some asymptotic expansion in $g$ as $g \to 0$ than $f(g) + e^{-1/g^2}$ has exactly the same ...

1

Instantons are characterized by the winding number and a set of collective parameters (e.g. location of the centers of the instantons, their sizes and the inequivalent orientations in the global group space / space-time). Quantum fluctuations of a unit winding number instanton can either leave the collective parameters unchanged (non-zero modes), or change ...

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