Is there something similar to Noether's theorem for discrete symmetries?

No, because discrete symmetries have no infinitesimal form which would give rise to the (characteristic of) conservation law. See also this article for a more detailed discussion.

You mentioned crystal symmetries. Crystals have a discrete translation invariance: It is not invariant under an infinitesimal translation, but invariant under translation by a lattice vector. The result of this is conservation of momentum up to a reciprocal lattice vector.

There is an additional result: Suppose the Hamiltonian itself is time independent, and suppose the symmetry is related to an operator $hat S$. An example would be the parity operator $hat P|xrangle = |-xrangle$. If this operator is a symmetry, then $[H,P] = 0$. But since the commutator of an operator with the Hamiltonian also gives you the derivative, you have $dot P = 0$.

Put into one sentence, Noether's first Theorem states that a continuous, global, off-shell symmetry of an action $S$ implies a local on-shell conservation law. By the words on-shell and off-shell are meant whether Euler-Lagrange equations of motion are satisfied or not.

Now the question asks if continuous can be replace by discrete?

It should immediately be stressed that Noether Theorem is a machine that for each input in form of an appropriate symmetry produces an output in form of a conservation law. To claim that a Noether Theorem is behind, it is not enough to just list a couple of pairs (symmetry, conservation law).

Now, where could a discrete version of Noether's Theorem live? A good bet is in a discrete lattice world, if one uses finite differences instead of differentiation. Let us investigate the situation.

Our intuitive idea is that finite symmetries, e.g., time reversal symmetry, etc, can not be used in a Noether Theorem in a lattice world because they don't work in a continuous world. Instead we pin our hopes to that discrete infinite symmetries that become continuous symmetries when the lattice spacings go to zero, can be used.

Imagine for simplicity a 1D point particle that can only be at discrete positions $q_tinmathbb{Z}a$ on a 1D lattice $mathbb{Z}a$ with lattice spacing $a$, and that time $tinmathbb{Z}$ is discrete as well. (This was, e.g., studied in J.C. Baez and J.M. Gilliam, Lett. Math. Phys. 31 (1994) 205; hat tip: Edward.) The velocity is the finite difference

$$v_{t+frac{1}{2}}:=q_{t+1}-q_tinmathbb{Z}a,$$

and is discrete as well. The action $S$ is

$$S[q]=sum_t L_t$$

with Lagrangian $L_t$ on the form

$$L_t=L_t(q_t,v_{t+frac{1}{2}}).$$

Define momentum $p_{t+frac{1}{2}}$ as

$$ p_{t+frac{1}{2}} := frac{partial L_t}{partial v_{t+frac{1}{2}}}. $$

Naively, the action $S$ should be extremized wrt. neighboring virtual discrete paths $q:mathbb{Z} tomathbb{Z}a$ to find the equation of motion. However, it does not seem feasible to extract a discrete Euler-Lagrange equation in this way, basically because it is not enough to Taylor expand to the first order in the variation $Delta q$ when the variation $Delta qinmathbb{Z}a$ is not infinitesimal. At this point, we throw our hands in the air, and declare that the virtual path $q+Delta q$ (as opposed to the stationary path $q$) does not have to lie in the lattice, but that it is free to take continuous values in $mathbb{R}$. We can now perform an infinitesimal variation without worrying about higher order contributions,

$$0 =delta S := S[q+delta q] - S[q] = sum_t left[frac{partial L_t}{partial q_t} delta q_t + p_{t+frac{1}{2}}delta v_{t+frac{1}{2}} right] $$ $$ =sum_t left[frac{partial L_t}{partial q_t} delta q_{t} + p_{t+frac{1}{2}}(delta q_{t+1}- delta q_t)right] $$ $$=sum_t left[frac{partial L_t}{partial q_t} - p_{t+frac{1}{2}} + p_{t-frac{1}{2}}right]delta q_t + sum_t left[p_{t+frac{1}{2}}delta q_{t+1}-p_{t-frac{1}{2}}delta q_t right].$$

Note that the last sum is telescopic. This implies (with suitable boundary conditions) the discrete Euler-Lagrange equation

$$frac{partial L_t}{partial q_t} = p_{t+frac{1}{2}}-p_{t-frac{1}{2}}.$$

This is the evolution equation. At this point it is not clear whether a solution for $q:mathbb{Z}tomathbb{R}$ will remain on the lattice $mathbb{Z}a$ if we specify two initial values on the lattice. We shall from now on restrict our considerations to such systems for consistency.

As an example, one may imagine that $q_t$ is a cyclic variable, i.e., that $L_t$ does not depend on $q_t$. We therefore have a discrete global translation symmetry $Delta q_t=a$. The Noether current is the momentum $p_{t+frac{1}{2}}$, and the Noether conservation law is that momentum $p_{t+frac{1}{2}}$ is conserved. This is certainly a nice observation. But this does not necessarily mean that a Noether Theorem is behind.

Imagine that the enemy has given us a global vertical symmetry $Delta q_t = Y(q_t)inmathbb{Z}a$, where $Y$ is an arbitrary function. (The words vertical and horizontal refer to translation in the $q$ direction and the $t$ direction, respectively. We will for simplicity not discuss symmetries with horizontal components.) The obvious candidate for the bare Noether current is

$$j_t = p_{t-frac{1}{2}}Y(q_t).$$

But it is unlikely that we would be able to prove that $j_t$ is conserved merely from the symmetry $0=S[q+Delta q] - S[q]$, which would now unavoidably involve higher order contributions. So while we stop short of declaring a no-go theorem, it certainly does not look promising.

Perhaps, we would be more successful if we only discretize time, and leave the coordinate space continuous? I might return with an update about this in the future.

An example from the continuous world that may be good to keep in mind: Consider a simple gravity pendulum with Lagrangian

$$L(varphi,dot{varphi}) = frac{m}{2}ell^2 dot{varphi}^2 + mgellcos(varphi).$$

It has a global discrete periodic symmetry $varphitovarphi+2pi$, but the (angular) momentum $p_{varphi}:=frac{partial L}{partialdot{varphi}}= mell^2dot{varphi}$ is not conserved if $gneq 0$.

Maybe,

http://www.technologyreview.com/blog/arxiv/26580/

I am by no means an expert, but I read this a few weeks ago. In that paper they consider a 2d lattice and construct an energy analogue. They show it behaves as energy should, and then conclude that for this energy to be conserved space-time would need to be invariant.

As was said before, this depends on what kind of 'discrete' symmetry you have: if you have a bona fide discrete symmetry, as e.g. $mathbb{Z}_n$, then the answer is in the negative in the context of Nöther's theorem(s) — even though there are conclusions that you can draw, as Moshe R. explained.

However, if you're talking about a discretized symmetry, i.e. a continuous symmetry (global or local) that has been somehow discretized, then you do have an analogue to Nöther's theorem(s) à la Regge calculus. A good talk introducing some of these concepts is Discrete Differential Forms, Gauge Theory, and Regge Calculus (PDF): the bottom line is that you have to find a Finite Difference Scheme that preserves your differential (and/or gauge) structure.

There's a big literature on Finite Difference Schemes for Differential Equations (ordinary and partial).

Sobering thoughts:

Conservation laws are not related to any symmetry, to tell the truth. For a mechanical system with N degrees of freedom there always are N conserved quantities. They are complicated combinations of the dynamical variables. Their existence is provided with existence of the problem solutions.

When there is a symmetry, the conserved quantities get just a simpler look.

EDIT: I do not know how they teach you but the conservation laws are not related to Noether theorem. The latter just shows how to construct some of conserved quantities from the problem Lagrangian and the problem solutions. Any combination of conserved quantities is also a conserved quantity. So what Noether gives is not unique at all.

Electric charge conservation is a "discrete" symmetry. Quarks and anti-quarks have discrete fractional electric charges (±1/3, ±2/3) electrons, positrons and protons have integer charges.

Actually there are analogies or generalisations of results which reduce to Noether's theorems under usual cases and which do hold for discrete (and not necesarily discretised) symmetries (including CPT-like symmetries)

For example see: Anthony C L Ashton (2008) Conservation Laws and Non-Lie Symmetries for Linear PDEs, Journal of Nonlinear Mathematical Physics, 15:3, 316-332, DOI: 10.2991/ jnmp.2008.15.3.5

Abstract We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.

The way followed is a succesive relaxation of the conditions of Noether's theorem on continuous (Lie) symmetries, which generalise the result in other cases.

For example (from above), emphasis, additions mine:

The connection between symmetry and conservation laws has been inherent in all of mathematical physics since Emmy Noether published, in 1918, her hugely influential work linking the two. ..[M]any have put forward approaches to study conservation laws, through a variety of different means. In each case, a conservation law is defined as follows.

Definition 1. Let $Delta[u] = 0$ be a system of equations depending on the independent variables $x = (x_1, dots , x_n)$, the dependent variables $u = (u_1, dots , u_m)$ and derivatives thereof. Then a conservation law for $Delta$ is defined by some $P = P[u]$ such that: $${operatorname{Div} P ; Big|}_{Delta=0} = 0 tag{1.1}$$

where $[u]$ denotes the coordinates on the $N$-th jet of $u$, with $N$ arbitrary.

Noether’s [original] theorem is applicable in the [special] case where $Delta[u] = 0$ arises as the Euler-Lagrange equation to an associated variational problem. It is well known that a PDE has a variational formulation if and only if it has self-adjoint Frechet derivative. That is to say: if the system of equations $Delta[u] = 0$ is such that $D_{Delta} = {D_{Delta}}^*$ then the following result is applicable.

Theorem (Noether). For a non-degenerate variational problem with $L[u] = int_{Omega} mathfrak{L} dx$, the correspondence between nontrivial equivalence classes of variational symmetries of $L[u]$ and nontrivial equivalence classes of conservation laws is one-to-one.

[..]Given that [the general set of symmetries] is far larger than those considered in the classical work of Noether, there is potentially an even stronger correspondence between symmetry and conservation laws for PDEs[..]

Definition 2. We say the operator $Gamma$ is a symmetry of the linear PDE $Delta[u] equiv L[u] = 0$ if there exists an operator $alpha_{Gamma}$ such that: $$[L, Gamma] = alpha_{Gamma} L$$ where $[cdot, cdot]$ denotes the commutator by composition of operators so $L Gamma = L circ Gamma$. We denote the set of all such symmetries by $sym(Delta)$.

Corollary 1. If $L$ is self-adjoint or skew-adjoint, then each $Gamma in sym(L)$ generates a conservation law.

Specificaly, for the Dirac Equation and CPT symmetry the following conservation law is derived (ibid.):

See:

• John David Logan, “First Integrals in the Discrete Variational Calculus,” Æquationes Mathematicæ 9, no. 2 (June 1, 1973): 210–20. DOI: 10.1007/BF01832628.

  The intent of this paper is to show that first integrals of the discrete Euler equation can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analog of the classical theorem of E. Noether in the Calculus of Variations.

If we can embed some discrete symmetry such as $mathbb{Z}/N$ via embedding to a continuous symmetry $U(1)$, then we can first derive the Noether theorem for the continuous symmetry $U(1)$. Next we can find the discretized conserved version of Noether current which should be conserved with values mod $N$.

It will be interesting to know whether this thought applies to discrete non-abelian symmetry by embedding to a non-abelian continuous symmetry group, and do the same procedure again.