48

Let's say space is really a lattice with spacing $\Delta x$. It turns out that this idea has more trouble with experiment than you might think, but we can plow ahead for the purposes of this question. You might propose replacing integrals in physics with discrete sums over individual lattice points, to take a concrete example let's think about the work ...


22

1) Gauge theory is a theory where we use more than one label to label the same quantum state. 2) Gauge “symmetry” is not a symmetry and can never be broken. This notion of gauge theory is quite unconventional, but true. When two different quantum states $|a\rangle$ and $|b\rangle$ (i.e. $\langle a|b\rangle=0$) have the same properties, we say that there ...


22

Lattice QCD calculations involve computing the inverse of the Dirac operator $\gamma\cdot D+m$. The difficulty of inverting an operator is controlled by its smallest eigenvalues, and computing the inverse of the Dirac operator becomes harder as $am\to 0$. The exact scaling of the computational cost depends on the algorithms. It was once feared that realistic ...


14

In the noninteracting case, the Hilbert space appropriate for a gauge field theory of any spin is a Fock space over the 1-particle space of solutions of the classical free gauge field equations for the same spin. (For spin 1, the associated particles would be noninteracting gluons if these would exist.) This space is ghost-free. The differences for ...


14

This is a comment, as Andrew's answer is adequate for the problem. I want to point out , which is not clear in your question, the difference between mathematical modeling and the object modeled. When modeling an object mathematically one can use continuous variables by the function of mathematics. If the object modeled has discontinuities, the mathematics ...


11

This is a supplement to Thomas's answer from March. I don't yet have enough reputation to add a comment, so let me write something not quite so brief but hopefully still clear. Although certain lattice QCD calculations with physical masses are currently feasible (and underway), many projects still need to use heavier masses. This is particularly true of ...


11

The correlation length of the 2d Ising model has been computed explicitly. You can find the expression in the famous book by McCoy and Wu. Here's a plot of the inverse correlation length (i.e., $1/\xi$) at various temperatures, taken from this recent review paper: This is only to show the directional dependence, as the radial scale is not the same for all ...


10

The Hilbert space of a gauge theory is defined by BRST symmetry or to be precise, BRST cohomology. In the path integral formalism, it is necessary to introduce ghosts in order to fix the gauge for a non-abelian theory. This theory now contains states of negative norm, hence it is a pseudo-Hilbert space. The Lagrangian of this theory has an additional ...


8

Your question is very interesting. I would like to mention something along the line of your question, but perhaps from another viewpoint. Recently there are some better understanding along the thinking between (1)"whether a theory is free from anomaly (the anomaly matching condition satisfied)," (2)"whether the symmetry of a theory is on-site symmetry," ...


8

After thinking about it I must say it is not as simple as I thought it would be. The JW transformation on the transverse Ising model contains quite a few subtleties. So to proceed, 1) Take your ground state for ANY $h$ expressed in the spinless fermion language. I stress ANY because this condition is true always - it's not just for $h<1$. Now this is ...


8

Topological degeneracy is only defined in the thermodynamic limit on a closed manifold. The ground state degeneracy of a finite-sized system or on an open manifold is not "topological", and can not be called topological degeneracy. Considering your examples. (1) The ground state degeneracy is ill-defined with open boundary condition. Because there might be ...


8

I take a different stance than Qmechanic: bosonization is `simply' the continuum version of the Jordan-Wigner transformation. Of course Qmechanic is right in that field theories are much more subtle than lattices theories. Nevertheless, the fact that JW is so simple does not mean it is not relevant when thinking of bosonization, in fact the opposite holds: ...


8

The critical point is not the same thing as the RG fixed point. Let $\mathcal{T}$ denote "theory space" meaning the set of all possible probability measures on real valued fields on the fixed unit lattice $\mathbb{Z}^2$. Block-spin or decimation etc. give you a map $R:\mathcal{T}\rightarrow \mathcal{T}$, namely, a renormalization group transformation. The ...


7

(This explanation is adapted from Nicholas Wheeler Notes, nevertheless is self-contained, also a slightly modified version is published on my website A Sudden Burst of Physics, Math and more ): I'll be using $a$ for the lattice spacing instead of $\Delta x$. One can clearly see how a quantity like $\mu = m/a$ (mass density per unit length) would yield a ...


7

I.1) Jordan-Wigner (JW) transformation. Let there be given a bosonic Heisenberg algebra of canonical commutation relations (CCR) $$[a_i,a_j^{\dagger}] ~=~\delta_{ij} {\bf 1}, \qquad [a_i,a_j] ~=~0, \qquad [a_i^{\dagger},a_j^{\dagger}] ~=~0, \qquad i,j~\in~\{1,\ldots, N\},\tag{1}$$ $$ n_i~\equiv~a^{\dagger}_i a_i \qquad\qquad\text{(no sum over $i$)}. \tag{...


7

The mean-field approximation amounts to evaluating the functional integral for the partition function in saddle point approximation, while the Gaussian approximation retains quadratic fluctuations around the saddle point and thus includes the lowest-order correction to the mean-field approximation in an expansion in fluctuations around the saddle point. ...


7

Yes, all lengths are multiples of the Planck length. All lengths are also multiples of the meter and the mile and the parsec. The Planck length is just another unit length and all lengths may be expressed as some multiple of any unit length. The Planck scale is expected to be the scale at which quantum gravity is expected to become important. That does not ...


6

I can answer (1) and (2). The answer is: NO. Passing form classical mechanics to quantum one requires, in general, to add more information. There is no rigorous machinery allowing one to write the quantum corresponding of a classical object. Physically speaking, this is because quantum structures are more fundamental in Nature than classical ones. ...


6

Honestly, I think this is one of those cases where you should just accept it and push on. This 'derivation' is really nothing more than a pedagogical device to make field theory seem somewhat natural to students with a background in classical mechanics. What we are trying to do is to take the continuum i.e. $N\to \infty$ limit of the following Lagrangian: ...


6

This is a great question. Pages 15 and 16 of these notes argues that no nontrivial spin Hamiltonian can ever be a fixed point under spin decimation, but I don't understand why their argument doesn't hold in the 1D case. The notes end with the cryptic comment there are many RG’s. The goal is not to see how many don’t work, but rather to find one that ...


6

$f(X)$ is a periodic function that has a Fourier series. In other words, $f(X)$ is a periodic function and so its Fourier transform has a spectrum at only discrete values of q. Still the transformed output $\tilde f(q)$ is potentially infinite and non-periodic in q-space. Electrons are not discrete in space but form a spread out cloud. It is assumed the ...


5

I have to admit that I do not know anything about the model you are working on, but the standard way to determine whether a gauge theory is confining or not is to calculate the vacuum expectation value of Wilson loops. The latter are gauge invariant operators that describe parallel transport around a closed loop in spacetime. If the vacuum expectation of a ...


5

The standard trick is partial bosonization, a.k.a. the "Hubbard-Stratonovich" trick. Consider ${\cal L}=g(\bar{\psi} \psi)^2$. Introduce a dummy field $\sigma$ with purely Gaussian lagrangian ${\cal L}_\sigma=-\frac{1}{g}\sigma^2$. You can always insert a factor 1 in the path integral $$ 1=\frac{1}{Z}\int D\sigma \exp(iS_\sigma). $$ Now shift the scalar ...


5

Edit after the discussion in the comments: Taking a look at this CEBAF paper, they rely on the lattice QCD calculations here which predict coverage of 110% on proton's spin. This graph in the latter demonstrates the relative components: $J^q$ stands for the total quarks' angular momentum and $J^g$ for the gluons' angular momentum (left). $J^q$ can be ...


5

Your last equation (v3) is not what Ryder meant. Think about it this way: The scalar field $\phi: \{1, \ldots, N\}^4 \to \mathbb{R}$ is a map from a discretized spacetime region $\{1, \ldots, N\}^4$ to a target space $\mathbb{R}$. For fixed lattice index $n\in \{1, \ldots, N\}^4$, the integration variable $\phi_n\in \mathbb{R}$ is a real number that ...


5

The answer seems to be: yes. For example, in an article by Hermele, Fisher and Balents (2003), they consider a $U(1)$ spin liquid in two dimensions where they concretely claim that ``[t]his state is stable to ALL zero-temperature perturbations''. It is however not clear to me whether this is a generic property of $U(1)$ spin liquids. Often people seem to ...


5

Here is a book chapter to solve your problem: Kopietz et al. “Mean-Field Theory and the Gaussian Approximation”. Lect. Notes Phys. 798, 23–52 (2010) [PDF].


5

I understand that any quantum lattice model at the critical point which can be described by a massless relativistic quantum field theory has emergent conformal invariance. This is not always true. For example, free quantum electrodynamics in $(2+1)$d is scale invariant and relativistic but not conformal. However, there is a proof that in $(1+1)$ dimensions, ...


5

In my opinion, one of the best modern references is a book by Gattringer and Lang https://www.springer.com/gp/book/9783642018497. This book contains rather a broad introduction of the subject, from the elementary details, such as path integral on lattice and different discretizations. And then there is discussion on more modern aspects, such as a various ...


5

"Tastes" is the name for the additional fermions produced by fermion doubling when putting actions with fermions on a lattice. "Taste symmetry" is a symmetry exchanging these additional fermions with each other. These "tastes" are unphysical and purely an artifact of the lattice theory - they have no relation to flavor except ...


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