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## Hot answers tagged mathematical-physics

18

I have personally done original research in the field of TQFTs, so I can tell you the reasons I find TQFTs interesting. Some that come to mind are: Some "real life" theories are accurately described/approximated by TQFTs; for example, gauge theories such as QCD. If you have a regular QFT that is gapped, then its strongly-coupled regime is almost ...

15

In condensed matter physics, topological quantum field theories provide an effective description of (many, but not all) gapped phases of matter at low energies and long distances. A phase of matter is gapped if it costs a finite amount of energy to create any excitation above the ground state. Examples of gapped phases of matter that admit a low-energy TQFT ...

4

When we compactify string theory, we are interested in the effective theory in the remaining dimensions, and perform a generalization of Kaluza-Klein reduction. Now, where in flat 10 dimensions the sum over worldsheets in the string amplitudes needed only to sum over the different kinds of worldsheets (their moduli and genus), we now have to distinguish ...

4

TQFTs were not discovered by mathematicians - they were actually discovered by physicists, so one should expect there to be physical motivation for the theory. One reason why that this is difficult to discover is that mathematicians have taken over the theory so it is hard to recognise the physical motivation. One reason to study them is as toy models for ...

4

Now by the last statement we should have a map from $\mathrm{SO}^+(1,3)$ to $\mathrm{SL}(2,\mathbb C)$. How can we define this map? That's not quite right. What that statement says is that we should have a map from paths$^\ddagger$ on $\mathrm{SO}^+(1,3)$ to paths on $\mathrm{SL}(2,\mathbb C)$. In other words, if we start at some point $p\in \mathrm{SO}^+(... 3 First, a misconception (or rather, an oversimplification). QFTs are local, but that does not mean they are insensitive to global properties. For example a free fermion with anti-periodic boundary conditions has a propagator$\psi(z)\psi(w)=1/(z-w)$but if you choose periodic boundary conditions, the propagator becomes$\psi(z)\psi(w)=(1/2)(\sqrt{z/w}+\sqrt{w/...

2

Producing the frequency spectrum of a time series is nothing but doing a Fourier transform, that is obtaining the amplitudes (and phases) of sinusoidal waves which result in the time series when all summed together. Given a time series $f(t)$ the Fourier amplitude in frequency space is $$F(\omega)=\int_{-\infty}^\infty dt f(t)\:e^{-i \omega t}$$ $F(\omega)$ ...

2

If you compacify to a torus then, in Cartesian coordinates, the spin connection vanishes and so is irrelevent. If you compactify to a sphere, as Fujikawa suggests, it is far less obvious that $\hat A(TM)$ is not needed. That it plays no role is because $\hat A(TM)$ is a genus and so cobordism invariant. This means that the curvature contribution to the ...

2

The heat kernel has four discrete indices in addition to the two continous space indices $x$,$y$ because the gamma matrices have two: $\gamma^{ab}$; and the matrix-valued gauge fields $A_\mu$'s that hide in the Dirac operator have two: $[A_\mu]_{kl}$. The heat kernel is therefore an operator acting on $L^2[{\mathbb R}^4]\otimes S\otimes V$ where $L^2[{... 2 Classically speaking, gauge fields are connections on principle$G$bundles. Matter fields, on the other hand, are sections of associated bundles. To construct an associated bundle, you must choose a representation of$G$. For instance, if$G = U(1)$, then this amounts to choosing an integer. When you construct the associated bundle, this integer is the ... 1 Both deal with theory rather than with experiments. The most striking difference is the level of rigour being put in. For example: In theoretical physics you solve differential equations, in mathematical physics you solve them as well, but you also prove that the solution exists and is unique or, if it is not unique, you consider all possible solutions and ... 1 In the axiomatic algebraic formalism, say of the Ostwalder-Schrader axioms of nets of local algebras, it turns that the local algebra must all be isomorphic to the unique hyperfinite factor of type$III_1$. Hence specific theories depend upon how these algebras embed within each other. For more details, see the paper The Role of Type III Facyors in QFT by ... 1 The answer by Jeanbaptiste Roux contains the exact form of the mapping (Lie group homomorphism) from$\mathrm{SL}(2,\mathbb C)$to$\mathrm{SO}^{+}(1,3;\mathbb{R})$(the latter is also denoted in the literature as$\mathfrak{L}^{+}_{\uparrow}$or$\mathfrak{Lor}(1,3)\$). As a mapping, it is not bijective, so it does not an inverse in the proper mathematical ...

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