## Wednesday, February 22, 2006

### Locality and additivity of energy

I found most of these debates about the "doubly special relativity" kind of crazy, so let me write a few general words about locality and additivity of energy.

Every physical theory that is supposed to describe our universe or any other remotely similar universe must satisfy the law of locality, at least approximately. What does it mean? It means that if you're doing something in Washington, DC, it should not directly influence events in Boston, MA. Well, as you know, Washington, DC may influence events in Massachusetts indirectly but it takes a certain amount of time for the influence to spread.

According to special relativity, you need time "t=s/c" where "s" is the distance and "c" is the speed of light. But even if you assume that the Lorentz invariance is not exactly satisfied in your setup, it should be clear that locality will continue to hold.

The laws of nature don't admit action at a distance, and if they do, such instantaneous effects must be very subtle and indirect. Note that we are assuming that objects can exist within space without modifying the asymptotic structure of spacetime which also means that we assume that the spacetime has 3+1 dimensions or more.

What does locality mean in our formalisms that describe the real world? Classical physics may be described in terms of a classical action or a classical Hamiltonian. If you study a system that is composed from two separated subsystems that are far from each other, the action or the Hamiltonian decompose into two pieces:
• S = S1 + S2
• H = H1 + H2

Consequently, if you impose the condition that the action is extremized and S1 only depends on some degrees of freedom and S2 only depends on other degrees of freedom, you will see that the extremality of S is equivalent to the extremality of S1 and the extremality of S2.

Analogously, if you derive the classical equations from the Hamiltonian H, the degrees of freedom of the first subsystem will only be affected by the terms in H1 while the other degrees of freedom will only be affected by H2. The subsystems decouple. We know that such a decoupling must be possible, at least approximately for large enough separations.

What about the quantum theory? At the quantum level, the Hilbert space Hil must contain states that look like tensor products of states from Hil1 and Hil2. The wavefunctions are products of wavefunctions of the subsystems. And if we assume that the two subsystems decouple, it implies that the Hamiltonian that generates time evolution must be equal to the sum of two individual Hamiltonians.

Do you prefer the path integral approach? You will only recover locality if the space of configurations of the whole system includes the Cartesian products of the histories of the subsystem 1 and those of the subsystem 2. Moreover, the action must again satisfy

• S = S1 + S2

which means that the phase in the path integral also factorizes

• exp(iS) = exp(iS1) exp(iS2)

which means that the calculated wavefunctions will again factorize into the products from the individual systems, much like the probabilities - which is the correct way how probabilities of independent events should behave.

As you can see, there must exist some sort of additivity of the action or the Hamiltonian. In relativistic field theory, the action is written as an integral over the whole spacetime and the Hamiltonian is written as an integral over the whole space and the requirements of locality are therefore trivially satisfied. This is also true at the quantum level assuming that the theory is anomaly-free and satisfies some other consistency criteria.

If you have a relativistic theory, the Hamiltonian itself becomes the time-component of a vector in spacetime. Not surprisingly, the remaining components of the vector - the momentum - can be written as local integrals, too. They also satisfy the constraint that the momentum of a composite system equals the sum of the momenta of the subsystems.

In BFSS matrix theory, the clustering property is realized slightly differently than in quantum field theory. Composite systems are described by block-diagonal matrices: each diagonal block encodes one subsystem. The contribution of the block off-diagonal modes, that are high-energy non-local degrees of freedom doomed to stay in their ground states, is zero. It is still true that the wavefunctions of composite systems factorize and the action and/or Hamiltonian (and momenta) are written as sums.

Whenever locality and separation of a system into pieces is a good approximation, the additivity of the action or energy-momentum and/or the multiplicativity of the path integral phase or the wavefunction must be satisfied.

In local theories with systems split into subsystems, the additivity holds not only for the action, the energy and the momentum but for any other additively conserved quantity, much like for the entropy.

The text above should make it clear that we cannot gain anything by a nonlinear redefinition of the energy or the momentum. For example, imagine that you use a "pseudoenergy" and "pseudomomentum" that differ from the usual energy and momentum by an overall factor that depends on the energy. You can immediately see that such pseudoenergy won't be additive. Non-linear function of energy is guaranteed to be non-additive. This also means that the redefinition of the energy for composite systems will differ from the formula redefining the energy for its individual parts. Locality will be violated and voodoo will become commonplace. If you wish, feel free to do research in the space of completely generic non-local theories but be ready that voodoo is not quite compatible with science.

The only way how you can save locality is that you actually find a reason why it is preserved. The only plausible way to prove locality is to show the equivalence of your theory with a theory where the energy and the momentum are additive. Redefined formulae for energy and momentum that fail to be additive are always inferior.

There is no point in using them. Formulae that may look natural in terms of non-linear functions of the the ordinary additive energy or momentum generically lead to non-local effects and the non-locality is only suppressed if it is manifestly suppressed in the ordinary, additive variables.

In the text above, I was describing subsystems that are truly separated. This requires the interactions to disappear. In the real world, it must be true that the interactions go to zero as the distance becomes large. At shorter distances, the interactions always matter.

In quantum field theory, one can still write down all interactions in terms of an action that is manifestly local because it is the integral of a local Lagrangian. In principle, quantum gravity may force you to think about more general theories where the action is not exactly an integral of a local quantity. The effective actions are nearly guaranteed to have non-local terms in them. But it is still true that at even longer distances, these non-localities must go away.

If there is no point in non-linear redefinitions of the energy and momentum, is there a reason why we should think about quantum deformation of the Poincare or de Sitter or anti de Sitter symmetries? My answer is mixed. In some contexts - like AdS/CFT or dS/CFT - we have at least some reasons to think that the quantum deformation of the symmetries could be relevant, especially because of the finite number of certain degrees of freedom in both cases. But as far as I know, no one has found a way to formulate dynamical rules - in terms of an action or its generalization - that enjoy a quantum deformation of a symmetry in 3+1 dimensions or higher. (There are solvable examples in two dimensions.) So this field remains speculative in nature. More obviously, proposing quantum-deformed theories without any justification of the deformation and without any idea how dynamics could be defined looks like a vacuous game to me.

Some time ago we discussed an unusual type of a star-product that was discussed in the context of three-dimensional theories. Such a construction would be obviously unphysical in 3+1 dimensions or higher because the additive rules for the momentum vector are violated, together with locality. In three dimensions it is plausible to imagine that a localized object always has nonlocal effects - because it creates a deficit angle - but in 3+1 dimensions or higher, such a reasoning is prohibited.

There are various directions in which our theories may be generalized but all of these generalizations are much more compatible with the principles of locality - and the associated analytical structure of the amplitudes. For example, perturbative string theory satisfies all analytical rules normally derived from local quantum field theories despite the fact that the elementary objects in string theory are extended.

In non-commutative field theory, the star product of exp(ipx) and exp(iqx) is still proportional to exp(i(p+q)x); just the overall phase may be affected by the non-commutativity. This is important for approximate locality represented by the additivity of the energy-momentum vector.

There are many modifications of special relativity and calculus in the literature that are extremely brutal in nature and it is very questionable whether mathematically ugly and unmotivated theories that drastically violate basic laws such as locality should be considered to be physics.