# Determinism, causality and field theory

1. Aug 10, 2013

hi
i read some text about causality and determinism, but i can't exactly distinguish between them.
what's really difference between them?

does not quantum mechanics respect one of them?

"Quantum ﬁeld theory includes causality as a fundamental axiom"
which one of axioms of quantum field theory due to causality?

2. Aug 10, 2013

### king vitamin

Determinism is the statement that, given an initial state, one can determine the state at a future time with certainty.

Causality is essentially a statement of cause and effect: a measurement in the future cannot effect a measurement in the past. In context of special relativity this must be a frame-independent statement, so one must say "space-like separated events must be completely uncorrelated." This is a technical statement meaning that two events at space-time locations (t,x,y,z) and (t',x',y',z') are totally uncorrelated whenever the inequality $c^2(t-t')^2 < (x-x')^2 + (y-y')^2 + (z-z')^2$ holds. As your source states, all relativistic quantum field theories must satisfy this.

3. Aug 10, 2013

### vanhees71

No, determinism means that all possible observables of a physical system take welldefined values. If you know the precise state of the system (e.g., the point in phase space of a system of point masses in classical mechanics or the electromagnetic field and charge-current distributions in classical electrodynamics). This doesn't say anything about the time evolution yet.

A theory is causal if (at least in principle, if the dynamical laws are precisely known) the state of the system is known for all times $t \leq t_0$ then also the state of the system at all later times $t>t_0$ is known (weak causality).

All fundamental theories of physics are causal in an even stronger sense. It's sufficient to know the precise state at one time $t_0$ to know the system's state at any later time $t>t_0$ (provided the dynamical laws are precisely known).

Quantum theory is causal but not deterministic. A complete determination of a state, i.e., the knowledge of an representing state vector $|\psi \rangle$ at time $t_0$, and the knowledge of the Hamiltonian of the system implies the knowledge of the state at any later time $t>t_0$. However, never all observables take determined values, for any state. The observables are represented by self-adjoint operators in state space and only if the state of the system is an eigenstate of an observable's representing operator, the value of this observable is determined to the corresponding eigenvalue. Most other observables won't have a determined value, but the state ket only tells you the probability to measure a possible value of the observable (an eigenvalue of the representing operator). According to quantum theory it is impossible to know more than these probabilities, because the state of the system is completely determined when the state ket is known.

A very good explanation (without any mathematics!) about this properties of quantum theory can be found in J. Schwinger, Quantum Mechanics - Formalism for Atomistic Measurements, Springer.

4. Aug 10, 2013

### Jolb

I think vanhees is on the right track here, and king.vitamin has a few misleading points.

Causality has to do with the time ordering of causes and effects (and can be defined without respect to the speed of light--which is a relativistic definition for causality). If B is an effect of the cause A, then the principle of causality would state that A occurred no later than B. It is really a rather simple concept if you do not concern yourself with relativistic effects, whereby A could appear to occur before B in one reference frame, but the opposite may be true in another reference frame.

Determinism means that: given the state of a system at some time t, the system's state at some later time t' can be deduced with 100% certainty. Whether QM is deterministic is a subtle question. The system's state |ψ> in fact evolves deterministically under the Schrodinger equation when it is not being measured. A non-deterministic element enters when a measurement is performed on |ψ>. So in the absence of measurements, QM is deterministic--but of course any sort of experiment requires measurements to be performed. Whether or not this non-deterministic feature is "real" depends on your interpretation of QM--for example in the Many-Worlds Interpretation, we think of the theory as entirely deterministic (|ψ> always evolves deterministically according to the SE), with the stochastic nature of subjective experience being the perplexing part. On the other hand, the Copenhagen interpretation states that measurements actually cause non-deterministic things to happen to |ψ>, and this is similarly perplexing.

In quantum mechanics, we can do experiments which seem to indicate instantaneous causality between distant measurements--and this would violate the strict causality principle king.vitamin mentioned. (The principle of causality I stated above, though, would work fine.) But QM does obey causality in a non-local way.

Furthermore, even though there are instantaneous causes and effects in QM, they cannot be used to send messages between people. It turns out that in QM, the transmission of "information" from one person to another does in fact respect the relativistic principle of (local) causality that king.vitamin has given. So we can't use QM to do sneaky things like instantaneously sending the results of horse races to a friend so she can place a bet before the race results arrive by an EM signal.

Last edited: Aug 10, 2013
5. Aug 11, 2013

### king vitamin

I'm rather surprised that determinism is defined by vanhees in a way that does not involve time evolution - this does not involve any definition of determinism that coincides with either the philosophical definition or the definition in dynamical systems (see http://www.scholarpedia.org/article/Dynamical_systems ). If Schwinger has a superior definition in terms of quantum observables, then of course, I can yield to such a definition if it is useful. It just does not coincide with the definition of determinism as it has existed for many years in previous fields of physics.

In any case, I think Jolb has the wrong interpretation of my statement. When I said that spacelike measurements are uncorrelated, I meant that all observables evaluated at spacelike separate events satisfy $[\phi(x),\phi(x') ]=0$, that is, that measurement of an arbitrary casual operator at spacelike separated events cannot possibly have relative uncertainty. I did not mean to leave out long range order in my definition.

I believe my definition of causality coincides with that of Weinberg. If my definition of determinism does not coincide with the reader's conventions, please ignore it.