Are Maxwell's equations deterministic in the sense that e.g. if given free space with H and E defined for any point at time t0, then Maxwell's equations are sufficient to determine H and E for any t>t0?
Maxwell's equations alone are not sufficient to determine H and E for any observer at any time. You would have to apply special and general relativity. AM
In free space, Maxwell's Equations reduce to a wave equation for E and a wave equation for H. The solutions are guranteed to be unique, and exist for all time. The equations are even linear, so will not display chaotic behavior (so that prediction is not just theoretically possible, but somewhere in the ballpark of actually possible). But if we include free charges the system becomes extremely nonlinear.
Yes, they are deterministic- but they may not be sufficient as in the case of a chaotic system. It's no different than the 3-body celectial mechanics problem. And I beleive you do not need special relativity since ME are already Lorentz invariant. You may in fact need GR if you want to solve problems near massive bodies.
Yes, Maxwell equations are deterministic, but determinism is only valid for the average of the physical ensemble. If E_{instant} = <E> + E_{fluctuation} Determinism works only for the average <E>. Molecular noise is not determinist. For example, fluctuations f in the electric field measured in a macroscopic solution of ions (near equilibrium) is indeterminist, only average behavior <f> = 0 <f f'>= delta(t-t') can be predicted.
Thanks for the answers. I tend to like determinism :) I have another question, namely: gieven ab initio free space with E and H do Maxwell's allow for the formation of matter out of this form of energy (energy of the electromagnetic waves). Arguablly Einstein says E=mc^2, so they probably should. I know I'm probbaly asking questions in a way one with a more detailed knowledge of the theories would not but I'd be happy of any comments on them anyway.
As far as I know, the only theories in which particles can be created from energy are relativistic Quantum Field theories.
Then you like an unphysical issue :-) Determinism does not work. Maxwell equations, like the rest of classical physics, are for constant number of particles,.
Why wouldn't determinism work. As far as I know Newton as well as Einstein are both deterministic. The only physical theory which is not is Quantum theory and I read (heard, I'm no longer sure) somewhere that even this one can be put on a deterministic footing just that then it doesn't sound that natural or something along does lines.
The answer to your initial question is yes, but .... Don't forget that M's equations involve sources, which can have their own dynamics. Technically the combined source-field equations are, in classical circumstances, deterministic. All this means is that second order differential equations, partial included, are fully determined for all t by the system of equations together with initial conditions. (See most any text on E&M) Note that self-energy issues, put some of the determinism in doubt -- the self-energy problem has never been satisfactorily solved. One could, I suppose, add a source in which the number of particles is a function of, say E*E + B*B. But now the set of equations is horribly non-linear, and all bets are off. Purely free fields are trivially deterministic, as are particles with no forces involved.. Regards, Reilly Atkinson
If you think that Determinism is invalidated by Quantum Mechanics, then you should read about Bohmian Mechanics. Even besides the existence of the purely deterministic Bohmian interpretation of quantum mechanics, you should remember that no one suspects that the ill understood field of quantum mechanics is a description of true reality.
The failure of determinism goes beyond QM. moreover Bohm formulation is incomplete and non scientific, since claim unobserved things.
The most simple reply "because world is not determinism." Determinism is a phylosophical option cannot be shown from physics. But undeterminism can be proved. E.g. QM.
No one ever observed a keplarian orbit either (they can't be directly observed from the earth), and this doesn't stop Kepler's theory of orbits from being a scientific theory. How is Bohmian mechanics incomplete? Bohmian mechanics, like any theory, postulates the existence of theoretical entities (things that cannot be observed, but can be infered to explain observations).
No one ever observed a keplarian orbit either (they can't be directly observed from the earth), and this doesn't stop Kepler's theory of orbits from being a scientific theory. How is Bohmian mechanics incomplete? Bohmian mechanics, like any theory, postulates the existence of theoretical entities (things that cannot be observed, but can be infered to explain observations). The predictive power of QM doesn't prove that the world is indeterminate. Determinism says that given the current universe state, there is one and only one future universe-state. Knowing the initial wave function does not allow us calculate one and only one definite universe state, but perhaps the wave function is not a complete description of the state i.e. Hidden Variable theories.
Determinism only works in simple systems. Like Einstein theory applied to one idealized body system, Newtonian mechanics applied to celestial bodies, etc. When Newton mechanics is applied to condensed matter does not work fine and is generalized by Langevin type equations. Almost all of statistical mechanics is not determinist, thermodynamics is not, chemical kinetics, of course is not, etc. At higher levels, biochemistry, ecology, medicine, sociology, dynamics of human populations, etc. determinism fails completely. It is not true that Quantum mechanics can be rewritten in a deterministic form. In fact, nobody has newer developed any deterministic consistent formulation of quantum mechanics. The problem of some people is that does not understand probability and think that probabilities of QM are some kind of measure of ignorance of some fundamental underlying formulation, which is, by ad hoc definition in those theories, unobservable. As said, determinism is a philosophical option with no solid link with pure physics because is not a testable scientific hypothesis. Moreover, there are further difficulties with idea of determinism regarding human matters (free well, ethic, etc.) Einstein theory is deterministic just like approximation, when stochastic (random) components vanish. That determinism does not work is not a philosophical issue, it is a fact of science. The problem of physics is that begin focusing in simple systems where determinism work at first approximation. There is no a culture of determinism on chemistry for example. ***************************** Kepler orbits are observed (Kepler theory does not say that orbits are unobservable items) and the theory tested by direct measure of position of planets which follows from Newton gravitation. I already said why Bohm mechanics is incomplete. Any theory to be taken seriously may be based in observed things. Bohm postulates the existence of things cannot be measured, even in principle. No QM alone, but scientific experiments prove (in a strict scientific sense) that world is not determinist. Do you know some single experimental proof of that world is determinist? Of course, the wave function is not a complete description of the state. That is already known. Wave functions work only for pure quantum states, it does not work for quantum mixtures. Another aspect ignored by Bohm...
OK I might as well add some of my own thoughts. First of all, when I say I like determinism, it does not mean that I don't find theories, which are based on probabilities and are said to be undeterministic, interesting and useful if not beautiful. I just mean that I would prefer a theory at least in principle to allow for only one evolution of an observed system (even if the underlying technical difficulties don't enable us to have closed formed solutions - as for example in the three body problem of Newton - there the movement is determined (except for singularities) even though we can't have closed formed solutions as in the two body case). QM on the other hand is inherently undterministic (as far as I know) in the sense that gievn a system, its future development is not determined. I also don't think that scientific experiments (can) prove (even if only in the limited strict scientific sense) that the world is undeterministic. What they show is that on a certain level of precision of physical measurments we have theories, which equipt us with tools necessary for the calculations of certin probabilities, that agree with the observed data within the error of measurment. But that doesn't mean that the world need be inherently undeterministic, let alone that we cannot have a deterministic theory, which would describe it. This may easily be seen with the introduction of hidden variables (which in my view don't make the theory incomplete). The succesfullness of a theory in physics may be measured only by its ability to predict the results of measurments and not by a formalism one chooses to have for it. The latter is more a matter of taste (as long as it does not entail that the theory would become less predicitve or make it harder to produce results) as it is a consequence of the world around us.
This is the twentieth century dichotomy in physics: Physicist want their scientific models to help them "really know" what's happening, but they have all bought the party line which says "all that matters in physics are observables". I dare you to look at the following picture of a theoretical model that could NEVER be observed, even in principle, but can only ever be infered INDIRECTLY from routine observations. This model is accepted as a practical fact today: http://eos.uom.gr/~hatzifot/orbits-trans.gif Why don't you take a look at the rest of the universe? For thousands of years western astronomers held the doctrine that the heavens were immutable, unchanging. The combined astronomical observations of all the humans who ever lived before the 15th century, considering that Astronomy was as important a skill in those days as driving a car is today, tend to support determinism.
I'm lost. The motion of the planets is indeed observable, in principle and in reality, and has been for a great deal of time.
Think harder. The picture I showed you is a cartoon. This is important because you will not find a photograph of the solar system from that perspective. The cartoon is a theoretical model, it is impossible (in principle, considering optics) to actually observe the exact scene that that cartoon depicts. Yet the predictions of the model are correct, and we accept the existence of orbits as a fact even though no has ever observed them as depicted in that cartoon. The point is that we can infer the existence of things that we don't see, if it makes our world view more conceptually pleasant.