QM vs SR: The Paradox of Simultaneity in Particle States

[robheus]

Acc. to QM no two particles can be in the same state at the same position at the same time, but how does that reconcile with SR where two observers do not agree on simultaneity?
Do some observers see baryonic stuff violate this postulate?

 

[jfy4]

That is the discrepancy between QM and Rel. In QM there are things like Time ordering operations, and space-time points. But in Rel. the idea of events being ordered in time is relative, and there are space-time points. This is the problem in physics many people are trying to solve right now, how to unite these two remarkably successful theories even though they are both so philosophically different.

 

[robheus]

That is the discrepancy between QM and Rel. In QM there are things like Time ordering operations, and space-time points. But in Rel. the idea of events being ordered in time is relative, and there are space-time points. This is the problem in physics many people are trying to solve right now, how to unite these two remarkably successful theories even though they are both so philosophically different.

Is it possible to design an experiment that could show wether a phenomena which acc. to QM theory alone would be impossible (Pauli exclusion), but acc. to SR (based on different perspective on simultaneity because of different inertial frames of reference) is possible?

 

[dulrich]

Two observers will agree on whether the particles are at the same position at the same time. The relativity of simultaneity has to do with whether events separated in space are simultaneous or not.

 

[humanino]

Acc. to QM no two particles can be in the same state at the same position at the same time, but how does that reconcile with SR where two observers do not agree on simultaneity?

In early discussions of SR, we often refer to two observers disagreeing on the simultaneity of events at different points in space. But an event in space-time remains defined as the intersection of two light rays, and that is manifestly Lorentz invariant. Qccording to QM no two particles can be in the same position at the same time, that is consistent with SR already.

edit
dulrich already answered while I was posting !

 

[torquil]

That is the discrepancy between QM and Rel. In QM there are things like Time ordering operations, and space-time points. But in Rel. the idea of events being ordered in time is relative, and there are space-time points. This is the problem in physics many people are trying to solve right now, how to unite these two remarkably successful theories even though they are both so philosophically different.

These two theories were united long ago in what is called relativistic quantum field theory. It seems you are talking about a quantum theory of gravitation, which is something different.

 

[Glen Bartusch]

Admit it, guys: there's this continuing war between QM and SR. And let's face it: those who side in the QM camp just don't understand Special Relativity. :)

 

[Glen Bartusch]

These two theories were united long ago in what is called relativistic quantum field theory. It seems you are talking about a quantum theory of gravitation, which is something different.

Lol, and what's the difference between gravitational mass and inertial mass? According to Einstein, not a thing. Whether the motion be translational or linear, the two forms are one and the same. This equivalence is what makes it impossible to unite 'gravity' with the Standard Model, and likewise illustrates the band-aid approach to some solutions in the problem of particle interaction.
If 'relativity' is true, and if 'quantum field' theory is true, then why the title, "relativistic quantum field theory"? Why not just "field theory"?

 

[Glen Bartusch]

Two observers will agree on whether the particles are at the same position at the same time. The relativity of simultaneity has to do with whether events separated in space are simultaneous or not.

Assume the particles to which you refer are photons. Then indeed, one observer must necessarily observe a younger or older photon relative to the observation of the other observer.

 

[Ben Niehoff]

If 'relativity' is true, and if 'quantum field' theory is true, then why the title, "relativistic quantum field theory"? Why not just "field theory"?

Because we need to distinguish it from classical field theories (such as electromagnetism and GR), and non-relativistic quantum field theories (such as many condensed matter theories).

"Relativistic" means that the theory obeys Lorentz symmetry. "Quantum" means the theory implements some version of the Heisenberg algebra. "Field theory" means it is a theory of functions on spacetime, which are called "fields".

 

[jostpuur]

Being in the same state is not same thing as being in the same position. Eigenstates of Hamiltonians are never perfectly point like.

Could it be that exclusion principle could be used to construct some paradoxes, just like entanglement can be used? Apparent instantaneous action, and that stuff?

 

[jfy4]

These two theories were united long ago in what is called relativistic quantum field theory. It seems you are talking about a quantum theory of gravitation, which is something different.

Relativistic Quantum Field Theory is the covariant form of quantum field theory, not a union between relativity and QM. Relativistic Quantum Field Theory is not a union, it is an expression of QFT in new terms. Relativity is the whole comprehensive relative motion of bodies, what we call gravity now, and has yet to be united successfully with QM.

 

[ismaili]

Relativistic Quantum Field Theory is the covariant form of quantum field theory, not a union between relativity and QM. Relativistic Quantum Field Theory is not a union, it is an expression of QFT in new terms. Relativity is the whole comprehensive relative motion of bodies, what we call gravity now, and has yet to be united successfully with QM.

I guess there is some misunderstanding between some jargons.
Jfy4 is talking about the union of "general relativity" with quantum mechanics.
However, Ben is talking about the union of "special relativity" with quantum mechanics.

And I think special relativity has been married to quantum mechanics well already.
Quantum Field Theory actually is the union.
And the possible candidates to describe "quantum gravity", which is the union of general relativity with quantum mechanics, include string theory.
Weinberg even pointed out several heuristic argument why the meet of special relativity and quantum mechanics produces "Quantum Field Theory" in his book.

 

[haael]

Acc. to QM no two particles can be in the same state at the same position at the same time, but how does that reconcile with SR where two observers do not agree on simultaneity?

The answer is very easy.

In QM time is also uncertain. In SR you can have one observer that thinks the time was t_0 and the other observer that has time t_1. But QM says that both these numbers are just random, in reality you have (simplifying) t_0 +- \Delta t and t_1 +- \Delta t. If these ranges overlap, the time is the same.

 

[alxm]

The Pauli Principle is not at odds with Special Relativity in any way.

It's a consequence of Special Relativity!

 

[tom.stoer]

Relativistic quantum field theory (rel. QFT) is perfectly well defined ; in this context there is no problem or clash of QM and SR.

The problem is that SR (and QFT) require a fixed spacetime background to set up the quantization procedure. Once this fixed stage goes away and you want to quantize on arbitrary dynamical spacetimes (as in GR!) it becomes difficult.

In addition time is not random in QM or QFT. It is a coordinate, not a dynamical observable (in QM and QFT observables are uncertain, coordinates and parameters are not).

 

[RUTA]

The Pauli Principle is not at odds with Special Relativity in any way.

It's a consequence of Special Relativity!

While the spacetime structure of QM is blockworld, it is not Minkowskian. See

G. Kaiser, J. Math. Phys. 22, 705-714 (1981) and
A. Bohr & O. Ulfbeck, Rev. Mod. Phys. 67, 1-35 (1995).

Specifically, as pointed out in Bohr & Ulfbeck (Eq. 76), the time coordinate transformation involves a translation (giving blockworld) but no factor of gamma (no dilation).

 

[robheus]

The Pauli Principle is not at odds with Special Relativity in any way.

It's a consequence of Special Relativity!

How?

 

[alxm]

How?

http://en.wikipedia.org/wiki/Spin-statistics_theorem"

While the spacetime structure of QM is blockworld, it is not Minkowskian.

And if my grandmother was a bus, she'd have wheels.

 

[BruceG]

In addition time is not random in QM or QFT. It is a coordinate, not a dynamical observable (in QM and QFT observables are uncertain, coordinates and parameters are not).

Quantum field theory textbooks are traditionally very bad at explaining this point. I found the introductory chapter in Quantum Field Theory [Srednicki] very illuminating. Regarding attempts at relativistic quantum mechanics:

"..We can solve our problem, but we must put space and time on an equal footing at the outset. There are two ways we can do this. One is to demote position from its status as an operator, and render it as an extra label, like time [this is the quantum field theory approach]. The other is to promote time to an operator..relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so."

The second approach though is used as standard in string theory, where space and time are operators parametrised by world sheet coordinates of the string.

 

[RUTA]

And if my grandmother was a bus, she'd have wheels.

Not necessarily. She could've been an electrical conductor. No wheels there.

 

[tom.stoer]

You certainly don't want to propose string theory here (as it procudes more problems than it solves.)

I think the first stept is to understand space and time in non-rel. qm., then in rel. qm, and after that to understand rel. QFT.

 

[BruceG]

You certainly don't want to propose string theory here (as it procudes more problems than it solves.)

I wasn't proposing string theory here as a solution to combining quantum mechanics and general relativity.

Sticking to special relativity, there are two ways to formulate a quantum theory consistent with Lorentz invaiance.

1. First Quantization approach: space-time is an operator.

2. Second Quantized approach: space-time parametises a quantum field.

Both ways are equally valid. Second Quantization is best for point particle theories because it keeps track of the possibly infinite number of particles. With String theory you can do more at the First Quantized level because the infinite particles are coded into the string vibrations. Of course there is no empirical justification for moving from points to strings.

 

[ismaili]

I wasn't proposing string theory here as a solution to combining quantum mechanics and general relativity.

Sticking to special relativity, there are two ways to formulate a quantum theory consistent with Lorentz invaiance.

1. First Quantization approach: space-time is an operator.

2. Second Quantized approach: space-time parametises a quantum field.

Both ways are equally valid. Second Quantization is best for point particle theories because it keeps track of the possibly infinite number of particles. With String theory you can do more at the First Quantized level because the infinite particles are coded into the string vibrations. Of course there is no empirical justification for moving from points to strings.

I think you proposed a good point.
But I don't quite understand the relation of this point with the first, and second quantisation.
Allow me to clarify what you were talking about.
Maybe I should ask the question: what is the definition of first and second quantisation?

My previous understanding is that, 2nd quantisation means we postulate equal-time canonical (anti-)commutation relations among fields; 1st quantisation means we are doing relativistic quantum mechanics only, i.e. we treat fields NOT as operators but as wavefuntions.
So, in this definition, first quantized theory should also treat space-time coordinates as parameters?

In string theory, we are actually doing relativistic quantum mechanical string.
Because we don't have fields of target space-time coordinates.
Instead, we have a 2D conformal field theory on the world-sheet.
But this doesn't mean that we have a quantum field theory of strings.
(I didn't study string field theory at all, so I don't know what's the quantum field theory of strings actually.)

So, in the first quantisation of strings, i.e. string theory, the target space-time coordinates become operators, which is unusual in first quantisation theories. Is this fact due to the connection of two faces of string theory:target/worldsheet space?
I mean, since we have to CFT on the worldsheet side, so all X^\mu(\tau,\sigma) become the quantized fields on the worldsheet, so X^\mu are operators.
Then, from the viewpoint of target space, we are doing first quantisation of strings, however, all space-time coordinates become operators.

Is my understanding correct?

 

[fundamentally]

The reason why relativity exists can only be understood in quaternionic Hilbert space. Relativity is caused by the way spacetime is defined. But the real cause is the quaternion waltz (c=ab/a) which equals b for complex numbers, but does not do so for quaternions. The waltz occurs when a unitary transform affects an observation. Thus, nearly always! With complex QM, you will never notice that it exists. The introduction of spacetime goes together with the introduction of proper time and coordinate time. If you describe dynamics by using coordinate time then you experience a Minkowski metric (or a Lorentzian metric in curved space). In that case exists a maximum speed c of information transfer. If you stay with proper time, then there is no maximum speed. The representation of an item in Hilbert space can move without being observed. In that case there is no speech of coordinate time. In that case only proper time makes sense.
More details can be found at http://www.scitech.nl/English/Science/Exampleproposition.pdf .

 

[BruceG]

Is my understanding correct?

Your understanding seems pretty much correct (at least as good as mine) - I probably wasn't very precise with my use of terms. As I understand it there are several equivalent ways to do quantization.

But just to clarify the one issue:

String theory, as normal done, is as you say "relativistic quantum mechanical string" and so only describes a single string. Properly string theory should be derived from a string field theory, but string interactions can be postulated directly just by considering different world sheet topologies e.g. a torus with 3 punctures gives a one-loop amplitude of three interacting strings.

To do the equivalent thing with point particles would be to do "relativistic quantum mechanical particle". So instead of your X^\mu(\tau,\sigma) you would just have X^u(tau). [sorry can't get Latex to work]

But then to extend this to an interacting theory you would have to have to keep track of arbitrary numbers of multiple particles corresponding to all the possible Feynman diagrams. This infinite nature of relativistic quantum mechanics is dealt with by changing instead to a quantum field (which is in effect an infinite number of quantum mechanical degrees of freedom parametrised by the space parameter).

You can see the way the different types of time (world line/sheet vs target space) are used in these different approaches but string theory and paritcle theory is not fundementally different on this aspect. Please ask someone cleverer than me to explain all this properly

 

[Fwiffo]

Acc. to QM no two particles can be in the same state at the same position at the same time, but how does that reconcile with SR where two observers do not agree on simultaneity?
Do some observers see baryonic stuff violate this postulate?

"Same position at the same time" is not right. Two fermions cannot be in the same quantum state!

 

[RUTA]

"Same position at the same time" is not right. Two fermions cannot be in the same quantum state!

Sure they can. In fact, Max Dresden asked this very question at a Chatauqua Short Course in ... 1995?

Supporse I have two hydrogen atoms on either side of the galaxy. The electrons in both atoms can be in the same quantum state. Suppose I bring these two atoms together and they form diatomic hydrogen. Now the electrons can't be in the same state. When does the Pauli exclusion principle take effect?

He didn't have an answer.

 

[tom.stoer]

Mathematically this is rather simple; whenever you try to construct a two-fermion quantum state like

a^\dagger_X a^\dagger_X |0> = 0

you get zero!

Here the operators are fermion creation operators and X labels all quantum state related properties like momentum, spin, isospin etc.

 

[RUTA]

Mathematically this is rather simple; whenever you try to construct a two-fermion quantum state like

a^\dagger_X a^\dagger_X |0> = 0

you get zero!

Here the operators are fermion creation operators and X labels all quantum state related properties like momentum, spin, isospin etc.

P1. The electron is a fermion.
P2. The electron is part of a hydrogen atom.
P3. There are many hydrogen atoms in their ground state at any given time in the universe.
P4. This ground state is a quantum state of the atom's electron.
C. There are many fermions in the same quantum state in the universe at any given time.

 

[xepma]

The Pauli Principle is not at odds with Special Relativity in any way.

It's a consequence of Special Relativity!

Going to be a bit nitpicky here, but the spin-statistics theorem is not a consequence of special relativity -- it's better to phrase it as: being compatible with special relativity. It's because you also have spin-statistics theorems for ordinary Euclidean space-time -- you don't need the full symmetry of special relativity as an input.

Sorry for nitpicking ;)

P1. The electron is a fermion.
P2. The electron is part of a hydrogen atom.
P3. There are many hydrogen atoms in their ground state at any given time in the universe.
P4. This ground state is a quantum state of the atom's electron.
C. There are many fermions in the same quantum state in the universe at any given time.

The quantum states belong to different hydrogen atoms -- ofcourse they are not the same quantum state. If you solve the problem of two well-seperated point-like hydrogen atoms you get a two-dimensional ground state sector in your Hilbert space.

Bring them together and either
a) you have interactions. These lift the degeneracy of the two states, forcing one to obtain a lower energy then the other. This is basics of band theory.
b) you have no interactions. The quantum states start to overlap and you deal with a degenerate system.

In both cases the electrons do not sit in the same quantum state.

This is basics stuff of solid state physics. Consult any book on solid state physics and they tell you how to deal with quantum states that partially overlap.

 

[tom.stoer]

In your example the quantum state is defined by

|X_i> = |nlms = 100\uparrow, R_i>

So R "distinguishes" the different electrons (in practice it's more difficult than that because one has to antisymmetrize the two-electron states, but the general argument should be clear)

 

[RUTA]

The quantum states belong to different hydrogen atoms -- ofcourse they are not the same quantum state. If you solve the problem of two well-seperated point-like hydrogen atoms you get a two-dimensional ground state sector in your Hilbert space.

Bring them together and either
a) you have interactions. These lift the degeneracy of the two states, forcing one to obtain a lower energy then the other. This is basics of band theory.
b) you have no interactions. The quantum states start to overlap and you deal with a degenerate system.

In both cases the electrons do not sit in the same quantum state.

This is basics stuff of solid state physics. Consult any book on solid state physics and they tell you how to deal with quantum states that partially overlap.

It's been some years since I took solid state physics. So, how close do they have to be?

 

[xepma]

It's been some years since I took solid state physics. So, how close do they have to be?

The Pauli principle is always there ofcourse. For it to be noticable you need to have an overlap between the two quantum states. There's no precise point at which the effect becomes noticable; it gradually increases. It's similar to the question at which distance chemical bonds start to form -- don't have a precise number for it, but it's probably some multiple of the Bohr radius, my guess would be.

 

[RUTA]

The Pauli principle is always there ofcourse. For it to be noticable you need to have an overlap between the two quantum states. There's no precise point at which the effect becomes noticable; it gradually increases. It's similar to the question at which distance chemical bonds start to form -- don't have a precise number for it, but it's probably some multiple of the Bohr radius, my guess would be.

Guess? I inferred from your advice to "consult any book on solid state physics" that the point at which one had to consider the interaction was well established. My bad.

 

[tom.stoer]

You need not consider interactions; formulated in terms of momentum space creation operators it becomes a purely algebraic exercise as you can see from my "calculation".

 

[RUTA]

You need not consider interactions; formulated in terms of momentum space creation operators it becomes a purely algebraic exercise as you can see from my "calculation".

But, if you want to get the new state for the two electrons, which previously were described independently of each other by one in the same algebraic system (you don't have a unique description for every fermion in the universe), something has to be changed. Right? So, how close to the atoms have to be to necessitate a new mathematical description for the pair as opposed to using the same description for the individuals?

 

[tom.stoer]

No, nothing changes. It's only the formulation in position space that is not very helpful as to create the |100+,R> state one has to use a complicated operator which is essentially an infinite product of elementary momentum space creation operators.

Think about a one-dimensional compact space (a circle) which results in discrete momentum space. As momentum is discrete all allowed states can be described using the following notation

|n_1^+ n_1^-, n_2^+ n_2^-, n_3^+ n_3^-, \ldots>

Here any n can take the values 0 or 1 and + and - refer to the spin quantum number.

Any allowed state can be written as

|10, 00, 11, \ldots>

You cannot identify one individual electron. You cannot describe any two electrons independently. They always know each other due to the Pauli principle; this principle does not care about there separation in position space.

a^\dagger_Y a^\dagger_X |0> = -a^\dagger_X a^\dagger_Y |0>

This guarantuees that whenever you try to construct a state where two electrons will be put in the same state you get something like

|\ldots, 2, \ldots> = 0

 

[RUTA]

No, nothing changes. It's only the formulation in position space that is not very helpful as to create the |100+,R> state one has to use a complicated operator which is essentially an infinite product of elementary momentum space creation operators.

Think about a one-dimensional compact space (a circle) which results in discrete momentum space. As momentum is discrete all allowed states can be described using the following notation

|n_1^+ n_1^-, n_2^+ n_2^-, n_3^+ n_3^-, \ldots>

Here any n can take the values 0 or 1 and + and - refer to the spin quantum number.

Any allowed state can be written as

|10, 00, 11, \ldots>

You cannot identify one individual electron. You cannot describe any two electrons independently. They always know each other due to the Pauli principle; this principle does not care about there separation in position space.

a^\dagger_Y a^\dagger_X |0> = -a^\dagger_X a^\dagger_Y |0>

This guarantuees that whenever you try to construct a state where two electrons will be put in the same state you get something like

|\ldots, 2, \ldots> = 0

What does this have to do with the energy levels of the electron in a hydrogen atom?

 

[Fwiffo]

But, if you want to get the new state for the two electrons, which previously were described independently of each other by one in the same algebraic system (you don't have a unique description for every fermion in the universe), something has to be changed. Right? So, how close to the atoms have to be to necessitate a new mathematical description for the pair as opposed to using the same description for the individuals?

I don't actually know the answer to this , but, the state includes all degrees of freedom, i.e space, momentum, spin etc... As long as you can tell your electrons aprart (the one on the right side and the one on the left side) there is no pauli principle. The pauli prinicple is in effect only when the electrons cannot be distinguished, i.e have some overlap, in phase space as well as all other degrees of freedom.
If we have one electron in a state |A> and another in a state |B> the "real" state would be |AB>-|BA>, We can ofcourse just re-lable our electorns as the electron in state B and the electron in state A, now we only need to be able to distinguish between them (i know the electron on the moon is on the moon and the one on Earth is on earth, but I have no way of knowing which one is electron number 1 and which is electron number 2).

In other words "good" fermion states are |AB>-|BA> or |AC>-|CA> but never |AA> or |BB>.

 

[Fwiffo]

Neutoron stars are the perfect example of how this whole thing works.

 

[tom.stoer]

What does this have to do with the energy levels of the electron in a hydrogen atom?

It is hard to explain how this works if you don't know the basics of qm.

What I have written down is a general state for some electron configuration in the universe. A state having on single "1" like |0,...,0,1,0,...> is a single electron state, a plane wave with fixed momentum k. Of course this does not apply directly to the state of an electron in a hydron atom. But using more complicated states you can construct all possible states including the |100> you are interested in.

The point is that the Pauli principle does not care about the specific details of the state. All what matters is that the algebraic construction takes care about never allowing two electrons being in the same state.

The antisymmetrization Fwiffo mentions is already build into this so called Fock space construction. You can use it to check how it works with wave functions. I will elaborate on that in a couple of minutes ...

 

[tom.stoer]

... back again.

From relativistic quantum field theory one can derive the so-called spin-statistics theorem; it says that wave functions of fermions (= spin 1/2) particles are antismmetric under exchange of any two particles.

In a two-particle state this will look as follows. Let the two particles with two different coordinates be described by two wave functions u_X(r_1) and v_Y(r_1-R) u and v determine the shape of the wave function, X and Y determine all other quantum numbers, in our case this is just spin.

Now let's assume that both electrons are in the 100 ground state but in different atoms lokated at different positions. That means u=v and X=Y. The antisymmetrized wave function of the two-electron system looks like

U(r_1, r_2) = u(r_1) u(r_2-R) - u(r_2) u(r_1-R)

Both vectors r₁ and r₂ are independent; they label what we would call the position of the individual particles. But in qm there are no individual particles; both particles are entangled, you are no longer allowed to say "electron A is here and electron B is there". All what you can say is "one electron is here the other one is there". There are no additional labels A and B. You see the difference?

Now if try to locate both particles at the same position, meaning you set R=0 you automatically get

U(r_1, r_2) = u(r_1) u(r_2) - u(r_2) u(r_1) = 0

That's basically my argument from above, now expressed in terms of wave functions instead of Focks states

 

[RUTA]

It is hard to explain how this works if you don't know the basics of qm.

What I have written down is a general state for some electron configuration in the universe. A state having on single "1" like |0,...,0,1,0,...> is a single electron state, a plane wave with fixed momentum k. Of course this does not apply directly to the state of an electron in a hydron atom. But using more complicated states you can construct all possible states including the |100> you are interested in.

The point is that the Pauli principle does not care about the specific details of the state. All what matters is that the algebraic construction takes care about never allowing two electrons being in the same state.

The antisymmetrization Fwiffo mentions is already build into this so called Fock space construction. You can use it to check how it works with wave functions. I will elaborate on that in a couple of minutes ...

I've taken and taught many courses on QM. I also just wrote a paper on the path integral interpretation of QFT, so I can probably keep up :-)

Your next post answers the question very well. I'll respond there, thanks.

 

[RUTA]

... back again.

From relativistic quantum field theory one can derive the so-called spin-statistics theorem; it says that wave functions of fermions (= spin 1/2) particles are antismmetric under exchange of any two particles.

In a two-particle state this will look as follows. Let the two particles with two different coordinates be described by two wave functions u_X(r_1) and v_Y(r_1-R) u and v determine the shape of the wave function, X and Y determine all other quantum numbers, in our case this is just spin.

Now let's assume that both electrons are in the 100 ground state but in different atoms lokated at different positions. That means u=v and X=Y. The antisymmetrized wave function of the two-electron system looks like

U(r_1, r_2) = u(r_1) u(r_2-R) - u(r_2) u(r_1-R)

Both vectors r₁ and r₂ are independent; they label what we would call the position of the individual particles. But in qm there are no individual particles; both particles are entangled, you are no longer allowed to say "electron A is here and electron B is there". All what you can say is "one electron is here the other one is there". There are no additional labels A and B. You see the difference?

Now if try to locate both particles at the same position, meaning you set R=0 you automatically get

U(r_1, r_2) = u(r_1) u(r_2) - u(r_2) u(r_1) = 0

That's basically my argument from above, now expressed in terms of wave functions instead of Focks states

Very nice, I can appreciate that you're not going to construct the Fock space representation for electrons in two hydrogen atoms -- your general approach suffices :-)

So, the question remains: When do I have to use the two particle entangled state as opposed to two single particle states? The answer can't be "always," because you'd have to put every fermion in the universe into every calculation.

 

[vanhees71]

So, the question remains: When do I have to use the two particle entangled state as opposed to two single particle states? The answer can't be "always," because you'd have to put every fermion in the universe into every calculation.

That's a good question. The answer is that fortunately, the particles are welldescribed by a local QFT which implies the linked-cluster theorem, according to which any experiment localized here and now is unaffected from unrelated other events in outer space. A very detailed analysis about this important feature of local QFT can be found in Weinberg, Quantum Theory of Fields, Vol. I.

 

[tom.stoer]

Very nice, I can appreciate that you're not going to construct the Fock space representation for electrons in two hydrogen atoms -- your general approach suffices :-)

So, the question remains: When do I have to use the two particle entangled state as opposed to two single particle states? The answer can't be "always," because you'd have to put every fermion in the universe into every calculation.

It depends on the states you want to describe. In principle you have to use the entangled state whenever there is "non-zero" overlap. So in principle this means "always".

The difference between Fock states and bound states known from qm (hydrogen atom) is that Fock states are constructed from plane waves which are not localized, whereas bound states are always localized. The question is not when one has to take all infinitly many electrons into account, but that using a Fock space construction requires always an infinite number of Fock states to construct a bound state. But that's not the question.

For bound states non-rel. qm is certainly enough. Then the approach is standard: use antisymmetrized states and start with a perturbation series to check if the cross-terms are relevant for your specific problem.

 

[alxm]

Going to be a bit nitpicky here, but the spin-statistics theorem is not a consequence of special relativity -- it's better to phrase it as: being compatible with special relativity. It's because you also have spin-statistics theorems for ordinary Euclidean space-time -- you don't need the full symmetry of special relativity as an input.

Well AFAIK, you need Lorentz invariance to prove the spin-statistics theorem. After which you're free to take the non-relativistic limit and show that the resulting antisymmetry relation still holds. So that's what I mean by 'consequence'.

It depends on the states you want to describe. In principle you have to use the entangled state whenever there is "non-zero" overlap. So in principle this means "always".

Or never.. (Since I recently wrote up a PF library entry on the Slater determinant ) If you describe your interacting electrons with non-interacting electronic states as a basis (configuration interaction), you can exactly describe your system as an expansion in terms of single-particle functions (with the caveat that a single 'orbital' function no longer represents the state of a single electron) So in that framework, the interaction energy can be related to the overlap integrals without any approximation being made.

In any case, the energy of the interaction between two atoms at a distance is pretty well known to die off as r^-6.

 

[tom.stoer]

In non-rel. qm no one "forces" you to use the antisymmetrized wave functions. It appears as a kind of interaction. Using rel. QFT you learn that it is a deeper principle which is automatically build in the Fock space approach but hidden in non-rel. qm. It is not an interaction.

That's why I started with the Fock space.

 

[alxm]

Well I didn't say it was an interaction, unless you interpreted 'configuration interaction' that way. It's a historical misnomer that stuck. Hartree himself disliked the name and always called it 'superposition of configurations', which is a better description.