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11

There are thousands of such examples, it is basically all situations in condensed matter physics. You see a lot of regularities that have no explanation. Here's one of the most annoying ones for me: Moseley's law--- you can knock out one of the two electrons most tightly bound to a heavy atom (in the K-shell). This leaves a hole orbiting the nucleus. The ...


10

A key difference between spontaneously broken symmetries and "emergent symmetries" is that emergent symmetries are never exact while spontaneously broken symmetries are backed by exact maths although the ground state isn't invariant. In most cases, the "emergent symmetries" only emerge if some parameters are fine-tuned, and even if it is so, they are only ...


9

First of all, unlike mass/energy, the entropy is not an observable. The word "observable" may be understood both as an adjective and as a noun ("an observable"). Entropy isn't an observable because it is not given by a linear operator acting on the Hilbert space (or on the space of density matrices), $$ L: |\psi\rangle \to L|\psi\rangle $$ Instead, the ...


8

At present, the Navier-Stokes equations for the dynamics of water haven't yet been derived from microscopic principles.


7

The superfluid phonons have the linear dispersion $\omega=ck$, and the low-energy effective theory is Lorentz invariant. So the superfluid is an example of emergent Lorentz symmetry. The statement: The velocity of sound in the superfluid is the same for all inertial observers, regardless of their relative motion to the superfluid is valid if the clock and ...


5

This is an example from hydrodynamics. When the effects of viscosity can be ignored (inviscid flow), a uniform incident flow can exert on immersed bodies only lift forces perpendicular to the asymptotic flow velocity. However, there exist an infinite number of solutions of the flow equations of motion satisfying the asymptotic conditions at infinity and the ...


4

Your second sentence is not correct in our current understanding of physics. Photons are not an "emergent property of space-time" - photons appear even when we take space-time to be a fixed object. Photons are "not due to electrons... jumping around" - they can be created by the motion of charges but they are independent objects. Photons are just as real as ...


4

The group of rotations of an $N$-dimensional space is $SO(N)$. Being a symmetry of nature, classical systems transform according to representations of $SO(N)$. Quantum mechanics, on the other hand, allows systems which transform according to the universal covering groups of classical symmetries. This is the reason why we get in three dimensional quantum ...


4

I think that there is two levels of answer in this question, whether we talk about an exact scheme (the RG is one in principle), or about the practical implementation/calculation. If one could implement the RG scheme exactly, one would capture emergence, since this is equivalent to solving the problem exactly. So, if you know the correct question, that is, ...


3

The simplest model is the spin-1/2 chain with Majumdar–Ghosh interaction: $$H=\sum_i P_{3/2}(i-1,i,i+1),$$ where $P_{3/2}(i,j,k)$ is the projection operator that projects a state onto the subspace with total spin-3/2 on sites $i,j,k$. The ground states are two dimer states (see the figure on wikipedia Majumdar–Ghosh model): ...


3

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 ...


3

It is curious that no one has answered the most obvious one: Non-perturvative Einstein(-Hilbert) general relativity.


2

Any problem that requires solving of non-trivial Schroedinger equations. For example, protein folding problem. It is known what equations the system should satisfy and those equations can be written down. Yet they cannot be solved with modern computers which would take millions of years tor that.


2

I was wondering something similar few month ago. Then I concluded that most of the topological staffs appear at the boundary between two different topological sector. A sector being characterised by a Chern number, or if you prefer a topological charge, one needs a boundary / an interface between two systems characterised by different topological charge. ...


2

As Ron noted, there are many, many examples within condensed matter; they often share a very similar story where the microscopic laws are known well (exactly, for the case of simulations), but the macroscopic laws are derived by symmetry concerns. Take for example, liquid crystals. We could simulate a collection of hard rods or ellipsoids - this is our ...


2

Perhaps let me try to address this question based on a discussion with Prof. Frank Wilczek. This post is not going to be complete or anything. The punch line question I discussed with him: is there a set of mathematical equations from physical principles to distinguish life and lifeless beings? (say, hand in a system as an input, one can check its live or ...


1

If you Google for something like complexity emergent properties you'll lots of interesting background reading on the subject. An executive summary is that physics is deterministic and therefore that all emergent properties are in principle predictable from a knowledge of the microsocpic behaviour. However in practice the complexity of the systems is so ...


1

Given a 1D universe containing a point that moves along a line, you can draw a 2D graph of position vs time. The graphs shows a fixed image of all time. But the ball is not fixed. It exists as a slice. The situation is similar in space-time. Yes, it is 4D. Yes our past, present, and future fill a block. But we only exist as a slice at the present. Space ...


1

In general, a low-energy effective theory inherits continuous symmetries of the microscopic model. For this reason, the effective theory of superfluidity of non-relativistic particles should necessarily be invariant under Galilean relativity. Galilean invariance was used long ago by people like London, Landau and Popov for construction of the effective ...


1

Another favorite: It's remarkably difficult to compute the nucleation rate of water molecules during a phase transation from microscopic equations. Water molecules have a dipole moment of order 1, so most of the usual approximation tricks don't work.


1

You have provided the von Neumann entropy definition which is derived from its density matrix. I would consider it an intrinsic rather than fundamental property, but this is just semantics. Some recent work by John Baez has investigated the dynamics of quantum entropy called quantropy.



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