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Why isn't momentum a function of position? 
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#19
Mar612, 04:09 PM

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Ilm 


#20
Mar612, 04:17 PM

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I suppose my question could be phrased in purely mathematical terms: under what conditions do the generators of a Lie algebra inherit the parametric dependence of the associated Lie group? 


#21
Mar612, 05:45 PM

P: 742

I don't think the answer is something special to quantum mechanics. Also in the classical Lagrangian/Hamiltonian formalism, conjugate momenta don't depend on their generalized coordinates. Maybe the question is similar to "why do we have second order differential equations?". Time is special, because it is not a conjugate quantity to anything or a function of such quantities. I.e. it's just a parameter and not an observable in Hamiltonian mechanics. 


#22
Mar612, 05:50 PM

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#23
Mar612, 06:18 PM

P: 742




#24
Mar612, 06:26 PM

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#25
Mar612, 06:46 PM

P: 742

For relativistic QM, there are physical reasons for the existence of spin. I really wonder what such reasons could be in the nonrelativistic case. 


#26
Mar612, 07:35 PM

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of the rotation group alone. Hegerfeldt, Kraus, Wigner, "Proof of the Fermion Superselection Rule without the Assumption of TimeReversal Invariance", J. Math. Phys., vol 9, no 12, (1968), p2029. Abstract: The superselection rule which separates states with integer angular momentum from those with halfinteger angular momentum is proved using only rotational invariance. Their argument is essentially a more rigorous version of the one in Ballentine's section 7.6 about rotations by ##2\pi##. This sort of thing can be developed to reveal a spinstatistics theorem for nonrelativistic QM. the parameters and then setting the parameters to 0. The confusion about time in dissipative Hamiltonian systems is a different issue. Let's go back to your original question: subsystem in some way. E.g., a dissipative system can gain or lose energy from/to another system, a system under the influence of a timedependent external force presumes the existence of another system responsible for that force, etc. So in general we have a composite system whose total Hamiltonian is timeindependent. But for the component subsystems, their individual evolution parameters might not coincide with a global time parameter associated with the total Hamiltonian. One chooses the componentspecific evolution parameters to make the maths as convenient as possible, and (presumably) to coincide with some notion of local clock associated with that subsystem. Herein lies a deep question about the distinction between kinematics and dynamics. There is a (noacceleration) theorem of CurrieJordanSudarshan which shows that assuming a common evolution parameter associated with interacting particles is not viable in general: their respective worldlines in a common Minkowski space fail to transform in a way which is compatible with the interacting versions of the Hamiltonian and Lorentz boost operators. The usual way to construct dynamics is the socalled "instant form" in which we add an interaction term to the Hamiltonian (and to the Lorentz boost generators in the relativistic case). This is motivated by our familiarity with our everyday picture of Euclidean 3D space and our imagined reference frame coordinatized implicitly in a way which is compatible with free dynamics. Ballentine describes this briefly on p83 where he justfies modifying only the Hamiltonian in the Galilean algebra to accommodate external fields. But this is not the only possible way that we can try to make a "split" between kinematics and dynamics. There's also the "point form" and "front form" which modify other generators, but I won't delve into the details here. It may even be the case that none of these relatively simple approaches are truly adequate for all purposes. Sudarshan and collaborators also experimented with more general alternatives in which the evolution parameter is determined dynamically rather than via a onceandforall split between kinematics and dynamics (which is what's done in the other forms of dynamics I listed above). I can probably dig out more references for the above if necessary, but that's probably enough for now. 


#27
Mar712, 02:30 PM

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In the same way spin is embedded in the Dirac equation, it is also embeded in a linearized version of the Schrödinger equation as shown by LevyLeblond and explained neatly in one of Greiner's books (either the one on symmetries, or the one on wave equations, don't remember which). 


#28
Mar712, 02:36 PM

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#29
Mar712, 02:48 PM

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#30
Mar712, 03:37 PM

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#31
Mar712, 03:40 PM

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#32
Mar712, 04:21 PM

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KleinGordon and Schrodinger's equations are linear. Surely you don't mean linearization but finding degree one equation.



#33
Mar712, 06:05 PM

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Yes, linearization = linear dependence of first derivatives.



#34
Mar712, 07:53 PM

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One can sometimes think more clearly about this stuff by reviewing some classical dynamics theory, and concentrating on canonical coordinates and momenta, and canonical transformations which preserve the dynamics (Hamilton's equations). E.g., if we start with a canonical pair (q,p) we can find transformations to a new pair (q'(q,p), p'(q,p)) such that canonical commutations relations (or Poisson brackets) still hold for (q',p'). But p' is a function of the old coordinate and momentum  this is not important, because we also have a new canonical coordinate q'. More generally, there are also "extended" canonical transformations which involve the time parameter. For more detail, try Goldstein, or Jose & Saletan. 


#35
Mar812, 01:18 AM

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#36
Mar812, 03:24 AM

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