Why isn't momentum a function of position?

[lugita15]

The procedure you found is almost correct, it could be made rigorous introducing a time-ordering:

<br /> U(t_1,t_2)= T\{exp{ {\large (} -\frac{i}{\hslash} \int_{t_1}^{t_2} \textbf{d}t \, H(t) {\large)} }\} \equiv <br /> \sum_{n=0...\infty} (-\frac{i}{\hslash})^n \frac{1}{n!} \int_{t_1}^{t_2} \textbf{d}\tau_1 \cdots \,\int_{t_1}^{t_2} \textbf{d}\tau_n \, T\{{H(\tau_1)\cdots H(\tau_n)} \}<br />

where T\{{H(\tau_1)\cdots H(\tau_n)} \} is the product obtained disposing hamiltonians evaluated in lower times to the right. Obviously if hamiltonians commutes at different times the time ordering has no effect.

Where can I find more information about the time-ordering? Is there any way to write this expression as an infinite product rather than an infinite sum?

If I interpreted it right I think we could reprase your question this way: could exist a representation of an abstract Lie group on a Hilbert space such that the representation of the generators of this group depends on the group parameter?

If this was the question I think the answer would be no, we can't. But I'm a bit confused thinking about the time dependent hamiltonian

I asked almost this exact question earlier this thread. I gather from naffin that the answer is that the family of time evolution operators does not in general form a one-parameter Lie group, so it doesn't have to be generated by a parameter-independent Lie algebra. So then the question becomes, why can't similar things occur for other operators?

 

[strangerep]

I think a good example would be one where the Hamiltonian operators at different times don't commute, but I'm afraid I can't think of one off the top of my head. Do you know of a good one?

Not really. I normally work with conservative systems. But in any case, it's likely to be simpler to try and understand the situation in the classical case first. Jose & Saletan give an example of a dissipative system with friction, but not in the Hamiltonian formalism. In such case, one must work with a semigroup (i.e., not necessarily having inverses), since such dissipative dynamics are not reversible.

For a time-dependent Hamiltonian in a conservative system, one can presumably find a canonical transformation to a new set of dynamical variables, and a new time variable $\tau$, such that the new Hamiltonian is independent of $\tau$.

why can't the spin angular momentum operator bear the same kind of nontrivial relationship to the intrinsic rotation group that a time-dependent Hamiltonion operator bears to the unitary time evolution group? The reason I raised this a bit earlier in this thread is that like the time evolution group and unlike the spatial translation group, the intrinsic rotation group has a parameter which does not correspond to an observable.

Even in the classical case, the dynamics of a rotating rigid body are far trickier than the ordinary cases. The configuration manifold is in fact the SO(3) matrices themselves, and the obvious choices for momenta -- the angular momentum generators -- cannot be used as canonical momenta in the usual Hamiltonian sense because they do not mutually commute in the Poisson bracket sense. (One can still write a Hamiltonian in terms of them, and an inertia vector, but this Hamiltonian is not expressed in terms of canonical momenta.) So one must work with other formalisms of dynamics if one wants to get anywhere.

BTW, I had another look through Goldstein and J+S. I now don't think Goldstein covers enough ground for this discussion. J+S cover more.

 

[Ilmrak]

Where can I find more information about the time-ordering? Is there any way to write this expression as an infinite product rather than an infinite sum?

I don't actually know where to find more on time ordering in QM, I would say any QM book as Sakurai, but I'm not sure. For QFT though, any book covers this (Peskin for example).

Anyway I'll give some sketch on how to find the expression I wrote.
Discretize the time between t_1 and t_2 dividing it in intervals \Delta t = (t_2 - t_1)/N. Now use the propagators in \Delta t, U(t_1 + k \Delta t , t_1 + (k+1)\Delta t) ≈ \exp{ \frac{-i}{ \hslash } H(t_1 + k \Delta t) \Delta t } ≈ 1 - \frac{i}{\hslash} H (t_1 + k \Delta t) \Delta t, to evolve the system in each time interval (each hamiltonian is commuting, at first order in \Delta t, with itself in the time interval it's propagating). The full propagator is the product of N of such propagators and it's naturally time-ordered. Now take the continuous limit and you're done.

I asked almost this exact question earlier this thread. I gather from naffin that the answer is that the family of time evolution operators does not in general form a one-parameter Lie group, so it doesn't have to be generated by a parameter-independent Lie algebra. So then the question becomes, why can't similar things occur for other operators?

Please tell me if this could be plausible.

There are transformation defined to represent a symmetry and transformations that instead makes sense for their own.

Time propagation is a transformation of the second type: it always makes sense. The system could be symmetric under time translation, so that time propagation forms a group, or it can be non conservative, so that time propagation is not a group. In the latter case hamiltonian could depend on the time.

Rotations in QM are instead defined to form a group. The definition of angular momentum do not depends on the system symmetries, it's always the same. If a system is rotationally invariant, then hamiltonian commutes with rotations. In QM angular momentum can't, by definition, depend on the angles.

This difference is because in QM there is only one real space dimension: the time. Coordinates are operators, they are more like fields then like space dimensions.

In QFT there are instead 3+1 spacetime dimensions. Rotation in the 3-dimensional space now makes sense fro their own, even when we write a non-rotational invariant hamiltonian.
So we can, in this case, find that angular momentum are "represented" (not in the group theory sense) on the fields with operators that depends on the angles and then do not forms a group.
We probably never see this case because we assume our theory is Lorentz invariant.

Ilm

 

[lugita15]

Anyway I'll give some sketch on how to find the expression I wrote.
Discretize the time between t_1 and t_2 dividing it in intervals \Delta t = (t_2 - t_1)/N. Now use the propagators in \Delta t, U(t_1 + k \Delta t , t_1 + (k+1)\Delta t) ≈ \exp{ \frac{-i}{ \hslash } H(t_1 + k \Delta t) \Delta t } ≈ 1 - \frac{i}{\hslash} H (t_1 + k \Delta t) \Delta t, to evolve the system in each time interval (each hamiltonian is commuting, at first order in \Delta t, with itself in the time interval it's propagating). The full propagator is the product of N of such propagators and it's naturally time-ordered. Now take the continuous limit and you're done.

How is the continuous limit of a product rigorously defined?

 

[strangerep]

Oh duh. I don't know why I didn't think of this before...

Some features of time-dependent Hamiltonians can be seen by studying the damped harmonic oscillator with external time-dependent applied force. Cf. J+S, sect 4.2.4. Goldstein also has some discussion of it, but J+S seems more extensive.) Instead of the usual solutions like $e^{i\omega t}$ one gets something like
$$
e^{(-\beta + i\omega) t}
$$
(plus a forced term which I haven't shown). The above can be interpreted in terms of a complex energy (where the imaginary part determines growth or decay of the oscillations).

An quantum example that's (sort of) related is the modeling of the formation and decay of unstable particles (resonances). Here, one uses a Hamiltonian which does not have strictly-real eigenvalues, so $e^{iHt}$ is not a unitary operator in the usual sense.

Instead of the usual state vector space, one uses a space of so-called Gamow vectors. Typically one needs a proper rigged Hilbert space, or a pair of Hardy spaces to represent all this satisfactorily. http://arxiv.org/abs/quant-ph/0201091

EDIT: Just found this recent paper:

Chandrasekar, Senthilvelan, Lakshmanan,
"On the Lagrangian and Hamiltonian description of the damped
linear harmonic oscillator",
Available as: http://arxiv.org/abs/nlin/0611048

Abstract:
Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

However, the Lagrangian and Hamiltonian they obtain are quite complicated -- see p7.

 

[Ilmrak]

How is the continuous limit of a product rigorously defined?

You essentially write that time-ordered product as a time ordered exponential of a sum, then take the limit for N \rightarrow \infty to obtain an integral from the sum.

I'm sorry but I can't find anything better then http://en.wikipedia.org/wiki/Path-ordering. If I should find something more complete I'll eventually post that reference.

Ilm

 

[Ilmrak]

Oh duh. I don't know why I didn't think of this before...

Some features of time-dependent Hamiltonians can be seen by studying the damped harmonic oscillator with external time-dependent applied force. Cf. J+S, sect 4.2.4. Goldstein also has some discussion of it, but J+S seems more extensive.) Instead of the usual solutions like $e^{i\omega t}$ one gets something like
$$
e^{(-\beta + i\omega) t}
$$
(plus a forced term which I haven't shown). The above can be interpreted in terms of a complex energy (where the imaginary part determines growth or decay of the oscillations).

An quantum example that's (sort of) related is the modeling of the formation and decay of unstable particles (resonances). Here, one uses a Hamiltonian which does not have strictly-real eigenvalues, so $e^{iHt}$ is not a unitary operator in the usual sense.

Instead of the usual state vector space, one uses a space of so-called Gamow vectors. Typically one needs a proper rigged Hilbert space, or a pair of Hardy spaces to represent all this satisfactorily. http://arxiv.org/abs/quant-ph/0201091

EDIT: Just found this recent paper:

Chandrasekar, Senthilvelan, Lakshmanan,
"On the Lagrangian and Hamiltonian description of the damped
linear harmonic oscillator",
Available as: http://arxiv.org/abs/nlin/0611048

Abstract:
Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

However, the Lagrangian and Hamiltonian they obtain are quite complicated -- see p7.

A non-unitary representation of the time evolution would describe a dissipative system, dissipation depending on the anti-hermitian component of energy.
The group structure would tough be preserved, so I suspect a system described by an hamiltonian not commuting with itself at different times would be something different and more complicated?

Also I never fully understood resonances. Time evolution for decaying particles, if I'm not wrong, is still unitary on the full Fock space, it's only its projection on the initial particle subspace that isn't unitary (is from anti hermitian component of the energy projection on the decaying particle subspace that resonance width come from?).

I'll try to read your references, hoping to understand something

Ilm

 

[lugita15]

You essentially write that time-ordered product as a time ordered exponential of a sum, then take the limit for N \rightarrow \infty to obtain an integral from the sum.

I'm sorry but I can't find anything better then http://en.wikipedia.org/wiki/Path-ordering. If I should find something more complete I'll eventually post that reference.

Ilmrak, have you found any references on how to properly define a continuous product? What is essentially required is some way to define a Type II product integral, except for general Lie algebras rather than just real numbers, where the exponential refers to the exponential map connecting the Lie algebra and the Lie group.

 

[lugita15]

Does anyone know anything further about the group structure known as the unitary propagator defined in the Reed and Simon excerpt in my post #51? Specifically, what can we say about the Lie algebra associated with a unitary propagator? Also, what can be said about the representation theory of a unitary propagator?

 

[Ilmrak]

Sorry, I didn't see your link to Reed and Simons...

The funny thing is that that article is a nice reference about the time-ordered expression for the propagator i wrote in post #58
Note in fact that the definition (X.129) is equivalent to mine in that post.
They then demonstrate that the operator U(t_1 , t_2) defined that way is exactly the propagator.

I think that's no need to warry too much on the continuous limit, simply take post #58 as a sloppy way do "derive" the right definition of the time evolution operator ^^

Ilm