Pauli exclusion principle: a Force or not?

[Zarqon]

A small part of my brain has been bugged by this for a while now, so I figured I'd ask. According to most teachings, as well as the wikipedia entry, there are only 4 fundamental forces. The strong and weak interaction, electromagnetism and gravity, and as far as I know the Pauli exclusion principle (PEP) has nothing to do with either one of those.

What's been bugging me is that for all practicle purposes the PEP seems to act like a force. For example, on the wiki page on forces it list the four fundamental forces, and then proceeds to list a number of forces that can all be dervied from these, but on several occasions throughout the text, e.g. while explaining the normal force, they say that it comes from the PEP, with no further link back to any of the four fundamental forces.

Further, in the case of a neutron star the situation is even more telling, namely that the size of the star is given simply by the relation of the gravity pressing matter inwards, and the fermi pressure, casused directly by the PEP, pressing matter outwards. This makes it seem very much like a force to me.


So, question: Why is the Pauli exclusion principle not the fifth fundamental force?

 

[Borek]

https://www.physicsforums.com/showthread.php?t=181298

 

[GeorgCantor]

Classical concepts are very often deficient for describing the quantum. As I stated in another thread:

"It is the immaterial principle that makes a thing to be what it is, not little atomic billiard balls. Think of it in terms of the exclusion principle of Pauli and the stability of matter. To really understand this, we have to learn to disregard our imagination."

Post 159 taken from:

https://www.physicsforums.com/showthread.php?t=396540&page=10

At the fundamental level, matter is almost entirely a set of relations, principles and rules.

 

[tom.stoer]

Looking at the mathematical implementation of the exclusion principle using creation operators it just says that

X = Y \to a^\dagger_X a^\dagger_Y |\ldots> = 0

That means that trying to "create" to identical fermions always gives zero. So there is no quantum mechanical state which can contain two identical fermions, because you are unable to create this state (even mathematically). Here X and Y label everything, all variables by which the two fermions could be distinguished: X = {momentum, spin, isospin, color, ...}

If you compare this rule for the field operators with expressions describing forces you will see that it's totally different.

 

[Zarqon]

Thanks for the links and comments, howerer, I'm afraid I'm not quite happy yet. As it seems to me that in the example with the neutron star, there is a force (gravity) getting canceled by something that comes from the PEP. This should imply that the two things are similar enough to have the same units at the very least, no?

If you compare this rule for the field operators with expressions describing forces you will see that it's totally different.

Well, I haven't gotten to QFT yet (am an experimentalist ) but as I just mentioned above, if they are so different, how come you can cancel a force by something coming from the PEP?

 

[unusualname]

If something can't be in a space because that space is occupied why is it a force? It's just a constraint on the possible configuration. A force is mediated by particle exchanges, the only doubt is gravity which is hypothesised to be a force mediated by gravitons which have yet to be observed, it may be that gravity is just a "classical" property of spacetime geometry, but this model doesn't seem to work at Planck scale.

 

[Borek]

When you put marbles in the box, and the box gets filled, why can't you put more marbles in the box? Is there a force involved, or not?

Honestly - I have no idea why the Pauli exclusion principle works the way it does, the only answer I know is "because that's the way it is". However, I understand where does the thread question comes from.

 

[DrDu]

The equilibrium of forces only holds in static equilibrium. However, the particles in a neutron star move with high velocity and it is rather a quantum mechanical extension of Newtons law (i.e. the acceleration ma and the mechanical forces are equal) which is relevant here. Similar things happen even in one-electron systems like, e.g., H_2^+ (Dihydrogen cation) where the Pauli principle is irrelevant. The forces holding the two nuclei together are the result of an equilibrium of nuclear repulsion, proton-electron attraction and the change of the kinetic energy of the electron with the distance of the protons. The latter "forces" are often called Hellmann-Feynman forces in this context.

 

[peteratcam]

Newton's equations define our perception of what a force is. Anything which doesn't look like it is in uniform motion has a force acting on it, our brains tell us. Well trained physicists will know that Newton's equations are only valid in an inertial frame in the classical, non-relativistic limit, where all particles are distinguishable though, and these days the word 'force' tends to mean particular interaction terms in a Lagrangian instead.

But since our perceptions are still basically informed by Newtonianism, we should answer the question from that perspective: almost any violation of Newton's equations of motion can be 'explained' by introducing arbitrary forces acting on particles. A famous example is centrifugal force.

In a rotating frame, Newton's equations are invalid, so we should just give up at trying to describe our experiences in that mathematical framework. We persist though, and after a bit of thought, we can conclude that we can still use Newton's equations to describe and understand our experiences (hurrah!), but only if we allow for a new force called the centrifugal force. The force is fictitious in the sense that it was invented purely for the purpose of being able to use Newton's equations. Luckily, the maths shows that motion in a rotating frame is exactly described by Newton's equations, as long as coriolis and centrifugal forces are included, so we don't feel so guiltly about applying Newton's equations outside of their realm of applicability.

Analogously, Newton's equations are not valid for describing the behaviour of quantum indistinguishable particles, so we should just give up (and use a fully quantum description). After a bit of thought, you can show that the behaviour of quantum particles can be described in Newtonian language, if we allow for a fictitious force called Pauli Exclusion, which is repulsive when electrons get too close. In this case, there is no exact equivalence: there is no modification of Newton's laws with arbitray forces that is the same as QM, but inventing a few forces like Pauli Exclusion gets you closer to an intuition about QM based on modified Newtonianism.

So if Pauli exclusion a force? No, it's just a constraint on allowed QM states, not a result of a potential energy term. Does it look, feel, and smell like a force? Yes, because it causes a violation of Newton I, and we are trained by experience to call every violation of Newton I a force.

 

[Academic]

When you put marbles in the box, and the box gets filled, why can't you put more marbles in the box? Is there a force involved, or not?

In that case I would say there is a force, the electromagnetic force.

 

[Zarqon]

Thanks for the long explanation peteratcam, it was helpful, and was infact what I was somewhat guessing. But this basically just means that the classical concept of "force" is somewhat outdated, and that it would have been better to explain violations of Newton's equations of motion using a more general class of fundamental "interactions" rather than only the four forces typically done today (like on the mentioned wiki articles). And that these could include not only things strictly defined as coming from the potential energy term, but everything that plays a force-like role.

 

[Cleonis]

[...] this basically just means that the classical concept of "force" is somewhat outdated, and that it would have been better to explain violations of Newton's equations of motion using a more general class of fundamental "interactions" rather than only the four forces typically done today (like on the mentioned wiki articles). And that these could include not only things strictly defined as coming from the potential energy term, but everything that plays a force-like role.

I agree.

It is common in the history of physics that when there are new developments the same words are still used but they are given new meaning. In relativistic physics the words 'space' and 'time' are still used, but there is a profound shift in meaning as compared to what the words 'space' and 'time' mean in classical context.

It would seem that the four fundamental forces are to be seen as members of a larger group. This larger group can be characterized as: factors that play a part in the outcome of physical processes.

It seems to me our current physics theories involve six such factors:
- Inertia,
- Gravitation,
- Electromagnetism,
- Pauli exclusion principle
- Strong nuclear force
- Weak nuclear force.

I think that currently the most notable unification is the scheme that unifies Electromagnetism, Strong nuclear force and Weak nuclear force.

Perhaps a successor to quantum physics will offer a unification of the Pauli exclusion principle with other fundamental factors.


By the way, I suppose there is another way of casting the difference:
In the case of the four fundamental forces it is physically possible to have a volume of space that is devoid of any presence of either of the four. (Or more precisely: the physics laws allow that you move arbitrarily far away from any force source.)

But there is no such thing as 'moving away from inertia', or 'moving away from the exclusion principle'. Inertia and the exclusion principle are always and everywhere.

 

[tom.stoer]

It is misleading to talk about forces as we know them from Newton; we should talk about interactions as they are defined in quantum field theory. I come back to the example I started with.

Let's assume we have a unique vacuum state from which we construct the space of all possible states, the so-called Fock space. The Fock space is a direct sum of all 0-, 1-, 2-, ... -particle Hilbert spaces. As I said above there must not be any state in tis space which contains two identical fermions. So in principle one can construct all possible states by considering all products of creation operators acting on the vacuum

\prod_n a^\dagger_{X_n} |0>

The Pauli principle then tells us that

X_m \neq X_n \;\forall\; m \neq n

otherwise the action of the product on the state produces 0 (not the vacuum state, just 0).

Up to now we have not introduced any interaction, so what I was saying is valid for a free theory in which no interaction takes place. If this is the case then the state I have just constructed remains invariant under time-evolution.

Example: if I create a state with one electron and one positron it stays exactly this: an electron-positron pair. They do not interact, they do not annihilate, they do not scatter, ... Nevertheless it is forbidden to have to identical electrons. A state with two identical electrons simply does not exist by construction (of course there's a deeper reason for the Pauli principle which we can discuss later).

Now we switch on the interaction. In quantum field theory that means that there is an operator which is constructed from at least three operators.As an example we could have something like

b^\dagger_{X_i} a_{X_m} a_{X_n}

Such an operator can e.g. destroy two fermions and create one photon. When acting on a state as defined above we either get zero (if there are no two electrons with matching properties which can be destroyed) or we get a new state with one photon. So an interaction transforms incoming states (e.g. containing two electrons) into outgoing states (e.g. with one photon).

Now we come back to the Pauli exclusion principle: It does not transform particles into other particles. It simply tells you on which states interactions can act (and which states are forbidden). It constrains the allowed states and the allowed interactions. An interaction forbidden by the Pauli principle is not an interaction which does nothing (like the identity operation), it is an interaction that is described by the 0-operator. It does not exist. It is not their.

Consider again a product like

a^\dagger_{X_n} a^\dagger_{X_n}

It annihilates all states in the theory. So not only is

a^\dagger_{X_n} a^\dagger_{X_n} |0> = 0

but even

a^\dagger_{X_n} a^\dagger_{X_n} = 0

The two-particle operator itself does not exist. It is zero.

So this means that the probability that something in your theory turns into two identical fermions vanishes. And this principle is valid for all theories containing fermions. It does not depend on the specific interaction.

So the Pauli principle is more fundamental than "forces" or "interactions"!

Let's make a simple example: consider a mathematical theory containing only real numbers. In this theory you must not calculate the square root of -1. This number simply does not exist. You can not act on it, you cannot multiply it, divide by it, nothing. So there is no mathematical operation acting on the square root of -1. This is not because the operation is not defined, it is because already the square root of -1 is not defined. There is nothing on which an operaton could act.

If you ask which operation (= "force") prevents imaginary numbers from being created, this is misleading. There is no force doing something with the imaginary numbers turning them into real numbers. There is no operation +, -, *, /, ... which is responsible for avoiding imaginary numbers. There is an underlying principle which from the very beginning exludes the existence of imaginary numbers.

 

[Cleonis]

So the Pauli principle is more fundamental than "forces" or "interactions"!

Let's make a simple example: consider a mathematical theory containing only real numbers. In this theory you must not calculate the square root of -1. This number simply does not exist. You can not act on it, you cannot multiply it, divide by it, nothing. So there is no mathematical operation acting on the square root of -1. This is not because the operation is not defined, it is because already the square root of -1 is not defined. There is nothing on which an operaton could act.

If you ask which operation (= "force") prevents imaginary numbers from being created, this is misleading. There is no force doing something with the imaginary numbers turning them into real numbers. There is no operation +, -, *, /, ... which is responsible for avoiding imaginary numbers. There is an underlying principle which from the very beginning exludes the existence of imaginary numbers.

This raises the question whether the metaphors 'electron degeneracy pressure' and 'neutron degeneracy pressure' should be in use at all. The word 'pressure' is quite evocative, but is it an appropriate word here?

What happens when a extinguished star collapses moves to the state of Neutron star? Let's say that for a while electron degeneracy pressure was just sufficient to prevent further collapse, but subsequently a large amount of matter accretes and the extuinguished star goes to further collapse.

In collapsing to a neutron star the matter itself transforms to another form. Normally that transformation is prohibitively unlikely. I suppose that in the case of collapse to neutron star the gravitational potential energy that is released by further contraction is larger than the energy cost of the matter transformation that is required.

Seen in that way the electron degeneracy pressure arises from the energy cost of the matter transformation that is required for further contraction.

 

[tom.stoer]

I think this is not a metaphore but due to a calculation for an thermodynamic quantity "pressure".

 

[DrDu]

I want to highlight the problem from another side. Consider a gas of noninteracting fermions. The Pauli principle then forces the particles to occupy excited one-particle states up to the Fermi energy. However, the way these particles exert forces on their surrounding is not fundamentally different from how classical particles do. Consider e.g. electrons confined to a trap consisting of two charges. The electrons will get reflected when approaching the charges but the way they exert force is of purely electromagnetic type.

 

[Zarqon]

I think this is not a metaphore but due to a calculation for an thermodynamic quantity "pressure".

This calculation may be the source of the confusion at least for the fermi gas example. Your description a few posts up is a good one concerning the micro-scale of a few fermions, and as you said, the PEP simply expresses that the probability is zero for two identical fermions. However, since I've read several times that the gravitational force in a neutron star really is balanced in some since by something coming for the PEP, I assume this to really be so, but what I'm really missing is how exactly.

What I mean is, it seems to me that you start with a probability (dimensionless), and then an integration is performed overall fermions in the neutron star, afterwhich you arrive at something that has the unit of Newton (because how can it otherwise cancel the gravity force?). And this intermediate integration is very unclear to me, exactly at what point does the PEP become a force?

On one place I saw an attempt at this explanation, and it basically stated that since the fermions in a neutron star are so tightly packed, i.e. their position is so well known, that their momentum is very unknown (from the HUP) and that this would lead to large momentum changes that on average would balance the gravity pressure. I don't know how true this explanation is though, but at least it gives a sense of units matching.

 

[tom.stoer]

I am no expert in calculations of neutron star properties, but I doubt that you will get something like force [Newton]. What you get is that the star does not collapse and you interpret this as a force.

Can you show me a reference where they do the math?

 

[Maaneli]

This calculation may be the source of the confusion at least for the fermi gas example. Your description a few posts up is a good one concerning the micro-scale of a few fermions, and as you said, the PEP simply expresses that the probability is zero for two identical fermions. However, since I've read several times that the gravitational force in a neutron star really is balanced in some since by something coming for the PEP, I assume this to really be so, but what I'm really missing is how exactly.

What I mean is, it seems to me that you start with a probability (dimensionless), and then an integration is performed overall fermions in the neutron star, afterwhich you arrive at something that has the unit of Newton (because how can it otherwise cancel the gravity force?). And this intermediate integration is very unclear to me, exactly at what point does the PEP become a force?

On one place I saw an attempt at this explanation, and it basically stated that since the fermions in a neutron star are so tightly packed, i.e. their position is so well known, that their momentum is very unknown (from the HUP) and that this would lead to large momentum changes that on average would balance the gravity pressure. I don't know how true this explanation is though, but at least it gives a sense of units matching.

Zarqon,

In spite of the responses you have been given thus far, I am here to tell you that your intuition about the PEP being a force, is in fact spot on! The de Broglie-Bohm formulation of QM explains electron degeneracy pressure in terms of a 'quantum force' (the gradient of the quantum potential) which, for example, in a collapsing neutron star, counter-balances the gravitational implosion. There was a rather extensive discussion about the quantum force explanation of the PEP in a previous thread. See for example (and make sure to read all the posts by zenith8):

https://www.physicsforums.com/showthread.php?t=364464

See also page 34 of this talk by Cambridge physicist Mike Towler:

http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/towler_pilot_waves.pdf

Feel free to let me know if you have further questions.

 

[Maaneli]

I am no expert in calculations of neutron star properties, but I doubt that you will get something like force [Newton]. What you get is that the star does not collapse and you interpret this as a force.

Can you show me a reference where they do the math?

See the second link in my post above.

 

[tom.stoer]

This summary of pilot wave theory is interesting in itself, but unfortunately it is not very explicit regarding spin and the Pauli principle (even in non-rel. qm), rel. qm / Dirac equation and especially regarding quantum field theory.

As it is not widely accepted I think we should try to understand the question regarding the Pauli pricniple and pressure based on the standard QM approach.

The Pauli principle is related to the spin-statistics-theorem which is deeply rooted in relativistic quantum field theory. I haven't seen any idea how pilot waves may work in relativistic quantum field theory.

 

[Maaneli]

As it is not widely accepted I think we should try to understand the question regarding the Pauli pricniple and pressure based on the standard QM approach.

The equations of the "standard" QM approach include all the equations of the de Broglie-Bohm approach. Hence, the quantum force picture applies even in the standard QM approach. You just need to rewrite the Schroedinger equation in the Madelung representation, and compute the resulting force from the quantum potential.

I haven't seen any idea how pilot waves may work in relativistic quantum field theory.

In fact it has been thoroughly developed in the past decade or so:

Field beables for quantum field theory
arXiv:0707.3685v2 [abstract, PS, PDF]

A minimalist pilot-wave model for quantum electrodynamics
with H. Westman
Proceedings of the Royal Society A 463, 3115-3129 (2007)
arXiv:0707.3487v2 [abstract, PS, PDF]

A Dirac sea pilot-wave model for quantum field theory
with S. Colin
Journal of Physics A: Mathematical and Theoretical 40, 7309-7341 (2007)
arXiv:quant-ph/0701085v2 [abstract, PS, PDF]

A new pilot-wave model for quantum field theory
with H. Westman
in Quantum Mechanics: Are there Quantum Jumps? and On the Present Status of Quantum Mechanics, eds. A. Bassi, D. Dürr, T. Weber and N. Zanghì, AIP Conference Proceedings 844, 321-339 (2006)
arXiv:quant-ph/0602229 [abstract, PS, PDF]

QFT as pilot-wave theory of particle creation and destruction
Authors: H. Nikolic
Comments: 29 pages, 2 figures, version accepted for publication in Int. J. Mod. Phys. A
Journal-ref: Int. J. Mod. Phys. A25:1477-1505, 2010

 

[Maaneli]

This summary of pilot wave theory is interesting in itself, but unfortunately it is not very explicit regarding spin and the Pauli principle (even in non-rel. qm), rel. qm / Dirac equation and especially regarding quantum field theory.

As it is a summary, it is not intended to be too explicit about such things. Nevertheless, see chapter 6 of this book for a more detailed discussion:

http://books.google.com/books?id=-H...AEwAQ#v=onepage&q=exclusion principle&f=false

 

[tom.stoer]

nevertheless we should come back to "Pauli exclusion principle: a Force or not?"

 

[zenith8]

[zenith awakes from a 6-month slumber..]

Hi Zarqon,

Maaneli was right to suggest Towler's Cambridge lectures to you. However, you might like to know there is a more recent one based on Rigg's book devoted entirely to this particular question, entitled "Exchange, antisymmetry and Pauli repulsion. Can we 'understand' or provide a physical basis for the Pauli Exclusion Principle?" which I found quite interesting. See http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/towler_pauli.pdf" .

Zenith8

 

[Maaneli]

[zenith awakes from a 6-month slumber..]

Hi Zarqon,

Maaneli was right to suggest Towler's Cambridge lectures to you. However, you might like to know there is a more recent one based on Rigg's book devoted entirely to this particular question, entitled "Exchange, antisymmetry and Pauli repulsion. Can we 'understand' or provide a physical basis for the Pauli Exclusion Principle? which I found quite interesting. See http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/towler_pauli.pdf" .

Zenith8

Thanks for posting this talk, I had not seen it yet.

 

[GeorgCantor]

What requires us to find a 'real' force behind the exclusion principle? Are the other fundamental forces 'real' at the quantum level? What happens to the 'real' electromagnetism and the strong nuclear force when the unobserved fullerene molecule passes through both slits at once? If this is just the desire to keep the Standard Model as it is(i.e. forces mediated by virtual particles), then the SM contradicts the double slit if we assume particles and forces are real at all times. So they are obviously not.

 

[tom.stoer]

I think the problem is due to the wording. If we are talking about "pressure" and "force" there should be something in the formulas allowing for this interpretation. The first problem is "how do these formulas look like?" whereas the second problem is if "force" and "pressure" are only macroscopic quantities or if even the microscopic formulas allow us to talk about "pressure" and "force".

My idea in the last posts was to show that what is called force or interaction is fundamentally different from the Pauli principle at the microscopic level , even if at the macroscopic level both effects seem to be pressure-like.

Nothing requires to find a real "force". Talking about macroscopic phenomena one feels rather comfortable, but at the fundamenatal quantum level it's missleading.

Nevertheless: does anybody know about a reference explaining the calculation for the neutron star?

 

[guerom00]

In a standard gas, let's say the air you breath, what causes the pressure ? It's the molecules velocity. What force would you associate with thermal pressure ?
The degenerate pressure is “the same” : as you compress a fermionic gas, the position of the individual particles becomes more and more sharp. Hence, according to HUP, their velocities becomes more and more broad. Hence the pressure…

 

[Maaneli]

I think the problem is due to the wording. If we are talking about "pressure" and "force" there should be something in the formulas allowing for this interpretation. The first problem is "how do these formulas look like?" whereas the second problem is if "force" and "pressure" are only macroscopic quantities or if even the microscopic formulas allow us to talk about "pressure" and "force".

My idea in the last posts was to show that what is called force or interaction is fundamentally different from the Pauli principle at the microscopic level , even if at the macroscopic level both effects seem to be pressure-like.

Nothing requires to find a real "force". Talking about macroscopic phenomena one feels rather comfortable, but at the fundamenatal quantum level it's missleading.

As I think Zenith and I have already pointed out (and which you seem to have oddly ignored), your idea is just wrong, since there is already a completely mathematically well-defined quantum mechanical definition of force that explains the PEP.

 

[alxm]

The degenerate pressure is “the same” : as you compress a fermionic gas, the position of the individual particles becomes more and more sharp. Hence, according to HUP, their velocities becomes more and more broad. Hence the pressure…

Actually that's not quite right. That holds true for both bosons and fermions; particle-in-a-box, shrink the box size and the K.E. goes up, true. It is an example of a boundary condition acting as 'pressure' though. (shall we call it "box pressure"?) But the bosons can exist in the same, lowest-energy state whereas the fermions are "pushed" into higher energy states, since the Pauli principle 'stops' them from being in the same state. So in addition to your "box pressure" you have the "degeneracy pressure".

On a sidenote, I've always found it a bit fascinating that astrophysics and chemical physics overlap a bit here. If you treat electrons in an atom as a homogenous gas (e.g. Thomas-Fermi model), which neglects exchange antisymmetry/Pauli principle, then you get analogous expressions for degeneracy pressure coming back as a correction.

 

[tom.stoer]

As I think Zenith and I have already pointed out (and which you seem to have oddly ignored), your idea is just wrong, since there is already a completely mathematically well-defined quantum mechanical definition of force that explains the PEP.

Calm down!

I do not oddly ignore the pilot wave theory. I just want to clarify what I already said in post #21: You try to lead us to believe in this pilot wave theory as an approach that is well established in quantum mechanics. This is simply wrong! It is not helpful to present this idea as the holy grail of quantum mechanics.

Please have a look at the title of the thread: Pauli exclusion principle: a Force or not? I just explained that the Pauli principle has nothing to do with forces (or interactions) as known from ordinary quantum field theory. Whereas numerous interactions can be constructed on top of the mathematical framework of (relativistic) quantum field theory the Pauli principle is itself part of this mathematical framework. Force, pressure etc. are secondary effects arising from a fundamental framework.

 

[Maaneli]

You try to lead us to believe in this pilot wave theory as an approach that is well established in quantum mechanics. This is simply wrong!

No, it is not wrong. The deBB theory is well established (particularly in foundations of QM, quantum chemistry, and AMO physics circles) as an empirically valid approach to QM. It is true that it it is not widely used or widely understood in the broader physics community; but the reasons for that have nothing to do with the empirical validity of the deBB theory. I suggest you do a little more research about deBB theory before making judgments about its current status in the physics community.

It is not helpful to present this idea as the holy grail of quantum mechanics.

I am not presenting it as the 'holy grail' of QM. That's your projection. I have simply pointed out that deBB provides the quantum force explanation of the PEP which Zarqon seeks. And this is something which you cannot get out of the textbook accounts of the PEP.

 

[Maaneli]

I just explained that the Pauli principle has nothing to do with forces (or interactions) as known from ordinary quantum field theory.

But this has already been explained multiple times by other people on this thread. So why repeat it?

Whereas numerous interactions can be constructed on top of the mathematical framework of (relativistic) quantum field theory the Pauli principle is itself part of this mathematical framework. Force, pressure etc. are secondary effects arising from a fundamental framework.

Yes, but now we can go beyond the ordinary framework of relativistic QFT, and get an account of the PEP in terms of a quantum force which is (arguably) fundamental.

Please have a read through the link to Towler's talk which Zenith8 posted.

 

[tom.stoer]

I still doubt your "werll established"; but let's forget about that.

... but now we can go beyond the ordinary framework of relativistic QFT, and get an account of the PEP in terms of a quantum force which is (arguably) fundamental.

Please have a read through the link to Towler's talk which Zenith8 posted.

I checked some of your references but I haven't found anything that is quantum field theory. Can one do S-matrix-, renromalization-group-, lattice-gauge-calculations etc.?

 

[Maaneli]

I still doubt your "werll established"; but let's forget about that.I checked some of your references but I haven't found anything that is quantum field theory. Can one do S-matrix-, renromalization-group-, lattice-gauge-calculations etc.?

You'll have to be more specific about which references you are referring to. But if you are referring to the pilot-wave field theory papers, and saying that you "haven't found anything that is 'quantum field theory'" in them, then I seriously question your base level knowledge and understanding of QFT (assuming you have in fact checked some of the references).

As for whether one can do S-matrix, renormalization group, and lattice-gauge calculations, the answers are yes, yes, and in-principle yes. I say 'in-principle' with regard to the last, because while there is no explicit pilot-wave formulation of lattice-gauge QCD, there do exists pilot-wave QFT models on a lattice, which reproduce the standard QFT predictions:

Bohmian Mechanics and Quantum Field Theory
Authors: Detlef Duerr, Sheldon Goldstein, Roderich Tumulka, Nino Zanghi
Journal reference: Phys.Rev.Lett. 93 (2004) 090402
http://arxiv.org/abs/quant-ph/0303156

Bell-Type Quantum Field Theories
Authors: Detlef Duerr, Sheldon Goldstein, Roderich Tumulka, Nino Zanghi
Journal reference: J.Phys. A38 (2005) R1
http://arxiv.org/abs/quant-ph/0407116

The continuum limit of the Bell model
Authors: Samuel Colin
http://arxiv.org/abs/quant-ph/0301119

A deterministic Bell model
Authors: Samuel Colin
Journal reference: Phys. Lett. A317 (2003), 349-358
http://arxiv.org/abs/quant-ph/0310055

Beables for Quantum Electrodynamics
Authors: Samuel Colin
To appear in the proceedings of the Peyresq conference on electromagnetism (September 2002). Annales de la Fondation de Broglie
http://arxiv.org/abs/quant-ph/0310056

Also, there is just no reason to think that there is any fundamental obstacle against specifically formulating a pilot-wave version of lattice-gauge theory.

 

[tom.stoer]

My impression is that these are still first attempts towards some toy models for QFT. So there is certainly not a fully developed Bohmian QFT available (but I admit that is to be expected as not so many people are working on Bohmian mechanics).

In addition it seems to me that one abandons determinism and introduces stochastic behaviour. I do not disagree that this may be the correct way to do it, but I question the whole approach towards a realistic, deterministic interpretation if one has to give up determinism even within Bohmian mechanics; what is the benefit of the whole approach then?

Last but not least I still do not see how the interpretation Pauli principle is affected. Even in these QFT-like papers one uses Grassman variables to deal with fermions. But in doing so one introduces the Pauli principe as a building principle into the theory w/o any explicit relation to "force" or "interaction".

So my claim from post #32 is still valid: the Pauli principle is itself part of this mathematical framework. Force, pressure etc. are secondary effects arising from it. Whereas numerous interactions can be given (not just the ones we observe in nature: electromagnetic force, strong force, ...), the Pauli principle is rooted in the structure of spacetime symmetry and has nothing to do with interactions constructed on top of it.

 

[Maaneli]

My impression is that these are still first attempts towards some toy models for QFT. So there is certainly not a fully developed Bohmian QFT available (but I admit that is to be expected as not so many people are working on Bohmian mechanics).

The one's by Duerr, Goldstein, Zanghi, and Tumulka (DGZT) fit that description. However, the models by Colin, Westman, and Struyve reproduces the full range of predictions of 'standard' abelian QFTs (though I'm not sure about the model by Nikolic). Although, I agree that they still seem to have somewhat of a 'cooked up' feel to them.

In addition it seems to me that one abandons determinism and introduces stochastic behaviour. I do not disagree that this may be the correct way to do it, but I question the whole approach towards a realistic, deterministic interpretation if one has to give up determinism even within Bohmian mechanics; what is the benefit of the whole approach then?

The use of stochastic behavior is only true of the DGZT Bell-type models. As Colin showed in his papers, one can take a continuum limit of those models, and get a fully deterministic fermionic pilot-wave QFT model. And as you can also see in Colin papers, he has generalized these deterministic models to reproduce the standard abelian QFT predictions. Also, the pilot-wave QFT models of Westman and Struyve are also deterministic. But it should be emphasized that the main goal of these pilot-wave QFT models is not just to restore determinism - rather, it is to supply QFT with a precise ontology, to solve the measurement problem while getting rid of the ad-hoc measurement postulates, and to even make testable new predictions for the case of quantum nonequilibrium in extreme astrophysical and cosmological situations:

Inflationary Cosmology as a Probe of Primordial Quantum Mechanics
Authors: Antony Valentini
http://arxiv.org/abs/0805.0163

De Broglie-Bohm Prediction of Quantum Violations for Cosmological Super-Hubble Modes
Authors: Antony Valentini
http://arxiv.org/abs/0804.4656

Black Holes, Information Loss, and Hidden Variables
Authors: Antony Valentini
http://arxiv.org/abs/hep-th/0407032

There are probably other good reasons for pursuing this pilot-wave approach to QFT, but the ones I cited are the most notables reasons, IMO.

Last but not least I still do not see how the interpretation Pauli principle is affected. Even in these QFT-like papers one uses Grassman variables to deal with fermions. But in doing so one introduces the Pauli principe as a building principle into the theory w/o any explicit relation to "force" or "interaction".

As Struyve points out in his paper 'Field Beables for Quantum Field Theory', one cannot use Grassman fields to make a pilot-wave QFT, since the Grassman fields do not seem to give a consistent probability interpretation. However, as Colin and Struyve show in their co-authored paper, using a Dirac sea pilot-wave model works just fine for fermions.

But no, one does not have to introduce the PEP as an additional, ad-hoc principle in these QFT models. If you write the wavefunctional in these pilot-wave QFT models in polar form, and separate the real and imaginary parts of its corresponding Schroedinger equation, you would get a QFT generalization of the quantum potential, and the corresponding quantum force. You can then explain the PEP in terms of the quantum force (particles get repelled away from nodes in the amplitude of the wavefunctional), just as you can with the quantum force in the first-quantized pilot-wave theories. So all of Towler's discussion (the talk that Zenith8 posted) of how to treat the PEP and symmetries of the fermionic wavefunction in first-quantized pilot-wave theory, can also be generalized to the pilot-wave QFT models of Colin, Westman, and Struyve.

So my claim from post #32 is still valid: the Pauli principle is itself part of this mathematical framework. Force, pressure etc. are secondary effects arising from it. Whereas numerous interactions can be given (not just the ones we observe in nature: electromagnetic force, strong force, ...), the Pauli principle is rooted in the structure of spacetime symmetry and has nothing to do with interactions constructed on top of it.

So no, your claim is not still valid.

 

[GeorgCantor]

You can then explain the PEP in terms of the quantum force (particles get repelled away from nodes in the amplitude of the wavefunctional), just as you can with the quantum force in the first-quantized pilot-wave theories.

Why doesn't this nodal repulsion force work in quantum tunneling?

 

[tom.stoer]

As Struyve points out in his paper 'Field Beables for Quantum Field Theory', one cannot use Grassman fields to make a pilot-wave QFT, since the Grassman fields do not seem to give a consistent probability interpretation. However, as Colin and Struyve show in their co-authored paper, using a Dirac sea pilot-wave model works just fine for fermions.

I do not see this "Colin-Struyve paper" in you last list. Can you add a reference?
And what is the conclusion for fermions, then?

But no, one does not have to introduce the PEP as an additional, ad-hoc principle in these QFT models.

It's not ad hoc. Asap you use Grassmann variables / anti-communiting field operators according to the spin statistics theorem it's natural.

If you write the wavefunctional in these pilot-wave QFT models in polar form, and separate the real and imaginary parts of its corresponding Schroedinger equation, you would get a QFT generalization of the quantum potential, and the corresponding quantum force. You can then explain the PEP in terms of the quantum force (particles get repelled away from nodes in the amplitude of the wavefunctional), just as you can with the quantum force in the first-quantized pilot-wave theories.

This sounds interesting in this context; to which paper are you referring to?

 

[peteratcam]

So my claim from post #32 is still valid: the Pauli principle is itself part of this mathematical framework. Force, pressure etc. are secondary effects arising from it. Whereas numerous interactions can be given (not just the ones we observe in nature: electromagnetic force, strong force, ...), the Pauli principle is rooted in the structure of spacetime symmetry and has nothing to do with interactions constructed on top of it.

So no, your claim is not still valid.

I have to weigh in on tom.stoer's side here. The pressure of a gas is a thermodynamic variable. To get the thermodynamics, you first do the statistical mechanics by doing a weighted sum over the allowed states of the degrees of freedom involved. The Pauli exclusion principle tells you what states to include in the sum (not how to weight them!), and then you derive the pressure from that. Discussing a mysterious "quantum force" is not relevant to the thermodynamic generalised forces like pressure.

As an aside, do the dBB crowd believe in eigenstates? Does quantum statistical mechanics work in the same way? I like magnets, and care less about particles - how should I think of the dBB theory of the hamiltonian H = J\mathbf S_1\cdot\mathbf S_2 ? I'm also a bit suspicious of the particle-centric dBB attempts at QFT. Surely it misses the point about a field theory

 

[Maaneli]

Why doesn't this nodal repulsion force work in quantum tunneling?

Why do you assume that it doesn't?

 

[Maaneli]

I have to weigh in on tom.stoer's side here. The pressure of a gas is a thermodynamic variable. To get the thermodynamics, you first do the statistical mechanics by doing a weighted sum over the allowed states of the degrees of freedom involved. The Pauli exclusion principle tells you what states to include in the sum (not how to weight them!), and then you derive the pressure from that. Discussing a mysterious "quantum force" is not relevant to the thermodynamic generalised forces like pressure.

First of all, the quantum force is not something 'mysterious'. Second of all, since the deBB theory gives you the dynamical equations for the trajectories of each particle (expressed in terms of the quantum force) in the gas, it can certainly give you the thermodynamic pressure as well. The claim here is not that the standard QM approach of using the PEP doesn't work - rather, the claim is that the standard QM approach has to take the PEP as a separate axiom of the theory with no further justification other than that it works, whereas the deBB approach to QM gives you a dynamical-causal explanation for the PEP.

As an aside, do the dBB crowd believe in eigenstates?

What do you mean by 'believe' in eigenstates? The wavefunction must certainly be regarded as an ontic field in deBB, if that's what you are asking about.

Does quantum statistical mechanics work in the same way?

Well no, not in the 'same way'. Because you have a law of motion for the particles, in addition to the Schroedinger evolution, the law of motion for the particles must also get modified for QSM. See for example:

Quantum dissipation in unbounded systems
Jeremy B. Maddox and Eric R. Bittner
Phys. Rev. E 65, 026143 (2002)
http://docs.google.com/viewer?a=v&q...c5x8M7&sig=AHIEtbQTvyTRy3ft07uVys5EfGnVGCqwmg

I like magnets, and care less about particles - how should I think of the dBB theory of the hamiltonian H = J\mathbf S_1\cdot\mathbf S_2 ?

In deBB theory, even your magnets are made up of particles. That Hamiltonian presumably has some corresponding wavefunction, in which case, there will necessarily be a corresponding continuity equation, and thus a corresponding guiding equation for the particles composing the magnets.

I'm also a bit suspicious of the particle-centric dBB attempts at QFT. Surely it misses the point about a field theory

First of all, there is nothing that a priori necessitates the deBB QFTs to be just like standard QFT. And in any case, those stochastic particle-centric deBB QFT's of DGZT do in fact reproduce the standard QFT predictions (for abelian gauge theories like QED and electroweak theory), as does the deterministic Dirac-sea pilot-wave QFT model of Colin and Struyve.

But if you want to insist on a pilot-wave theory in terms of field configurations only, you can have that too, as shown in the papers I referenced by Struyve and Westman.

 

[Maaneli]

I do not see this "Colin-Struyve paper" in you last list. Can you add a reference?
And what is the conclusion for fermions, then?

I referenced it in post #22. Here is the abstract:

"We present a pilot-wave model for quantum field theory in which the Dirac sea is taken seriously. The model ascribes particle trajectories to all the fermions, including the fermions filling the Dirac sea. The model is deterministic and applies to the regime in which fermion number is superselected. This work is a further elaboration of work by Colin, in which a Dirac sea pilot-wave model is presented for quantum electrodynamics. We extend his work to non-electromagnetic interactions, we discuss a cut-off regularization of the pilot-wave model and study how it reproduces the standard quantum predictions. The Dirac sea pilot-wave model can be seen as a possible continuum generalization of a lattice model by Bell. It can also be seen as a development and generalization of the ideas by Bohm, Hiley and Kaloyerou, who also suggested the use of the Dirac sea for the development of a pilot-wave model for quantum electrodynamics."
http://arxiv.org/abs/quant-ph/0701085

It's not ad hoc. Asap you use Grassmann variables / anti-communiting field operators according to the spin statistics theorem it's natural.

Yes, the spin-statistics theorem is ad-hoc. It merely postulates that fermionic wavefunctions are anti-symmetric under exchange of particle positions, and that bosonic wavefunctions are symmetric under exchange. Towler also explains this in his talk.

By contrast, in deBB, you can actually *derive* these postulates from the particle dynamics. A rigorous proof of this was given by Guido Bacciagaluppi in the context of the first-quantized deBB theory:

Derivation of the Symmetry Postulates for Identical Particles from Pilot-Wave Theories
Authors: Guido Bacciagaluppi
http://arxiv.org/abs/quant-ph/0302099

Remarks on identical particles in de Broglie-Bohm theory
Authors: Harvey R. Brown (Oxford), Erik Sjoeqvist (Uppsala), Guido Bacciagaluppi (Oxford)
Journal reference: Phys. Lett. A251 (1999) 229-235
http://arxiv.org/abs/quant-ph/9811054

This sounds interesting in this context; to which paper are you referring to?

This polar decomposition of the Schroedinger wavefunctional was primarily used by Bohm, Kaloyeraou, and Holland in their approaches to field theory. Struyve makes brief reference to it in his 'Field Beables' paper, but does not explicitly make use of it because he prefers the simpler first-order pilot-wave dynamics approach. Nevertheless, it is trivial to show that it is always possible to write the Schroedinger (or Klein-Gordon or Dirac) equation for the wavefunctional into a Madelung form with a quantum potential. Try it yourself! But if you would also like to see a concrete example of this being done (for the nonrelativistic Schroedinger case), have a look at page 519, section 12.4 "QFT in the Schroedinger picture and its interpretation" of Holland's book, The Quantum Theory of Motion:

http://books.google.com/books?id=Bs...AEwAw#v=onepage&q=schrodinger picture&f=false

 

[tom.stoer]

Yes, the spin-statistics theorem is ad-hoc. It merely postulates that fermionic wavefunctions are anti-symmetric under exchange of particle positions, and that bosonic wavefunctions are symmetric under exchange.

Let's look at Wikipedia http://en.wikipedia.org/wiki/Spin-statistics_theorem which I cite because I have no access to the original papers

The theorem states that:

• the wave function of a system of identical integer-spin particles has the same value when the positions of any two particles are swapped. Particles with wavefunctions symmetric under exchange are called bosons;
• the wave function of a system of identical half-integer spin particles changes sign when two particles are swapped. Particles with wavefunctions anti-symmetric under exchange are called fermions.

In other words, the spin-statistics theorem states that integer spin particles are bosons, while half-integer spin particles are fermions.

The spin-statistics relation was first formulated in 1939 by Markus Fierz,[1] and was rederived in a more systematic way by Wolfgang Pauli.[2] Fierz and Pauli argued by enumerating all free field theories, requiring that there should be quadratic forms for locally commuting observables including a positive definite energy density. A more conceptual argument was provided by Julian Schwinger in 1950. Richard Feynman gave a demonstration by demanding unitarity for scattering as an external potential is varied,[3] which when translated to field language is a condition on the quadratic operator that couples to the potential.[4]

Any proof of the theorem requires relativity, since the nonrelativistic Schrodinger field can be consistently formulated with any spin and either statistics.

I don't think that it's ad-hoc. It's not a postulate but a theorem.

 

[Maaneli]

Let's look at Wikipedia http://en.wikipedia.org/wiki/Spin-statistics_theorem which I cite because I have no access to the original papers



I don't think that it's ad-hoc. It's not a postulate but a theorem.

I did not say that the spin-statistics theorem is a postulate, I said the spin-statistics theorem postulates said symmetries of the wavefunction. In other words, those symmetries are merely taken as axioms of the theorem. And the theorem's 'proof' from relativistic invariance merely shows that these axioms are consistent with relativistic invariance. Towler also argues this:

"Often claimed antisymmetric form of fermionic Ψ arises from relativistic invariance requirement, i.e. it is conclusively established by the spin-statistics theorem of quantum field theory (Fierz 1939, Pauli 1940). Not so - relativistic invariance merely consistent with antisymmetric wave functions. Consider:

Postulate 1: Every type of particle is such that its aggregates can take only symmetric states (boson) or antisymmetric states (fermion).

All known particles are bosons or fermions. All known bosons have integer spin and all known fermions have half-integer spin. So there must be - and there is - a connection between statistics (i.e. symmetry of states) and spin. But what does Pauli’s proof actually establish?

• Non-integer-spin particles (fermions) cannot consistently be quantized with symmetrical states (i.e. field operators cannot obey boson commutation relationship)
• Integer-spin particles (bosons) cannot be quantized with antisymmetrical states (i.e. field operators cannot obey fermion commutation relationship).

Logically, this does not lead to Postulate 1 (even in relativistic QM). If particles with integer spin cannot be fermions, it does not follow that they are bosons, i.e. it does not follow that symmetrical/antisymmetrical states are the only possible ones (see e.g. ‘parastatistics’). Pauli’s result shows that if only symmetrical and antisymmetrical states possible, then non-integer-spin particles should be fermions and integer-spin particles bosons. But point at issue is whether the existence of only symmetrical and antisymmetrical states can be derived from some deeper principle.

Actually, fact that fermionic wave function is antisymmetric - rather than symmetric or some other symmetry or no symmetry at all - has not been satisfactorily explained. Additional postulate of orthodox QM. Furthermore, antisymmetry cannot be given physical explanation as wave function only considered to be an abstract entity that does not represent anything physically real."

Even Pauli himself recognized the ad hocness of his EP:

“..[the Exclusion Principle] remains an independent principle which excludes a class of mathematically possible solutions of the wave equation. .. the history of the Exclusion Principle is thus already an old one, but its conclusion has not yet been written. .. it is not possible to say beforehand where and when one can expect the further development..” [Pauli, 1946]

“ I was unable to give a logical reason for the Exclusion Principle or to deduce it from more general assumptions. .. in the beginning I hoped that the new quantum mechanics [would] also rigorously deduce the Exclusion Principle.” [Pauli, 1947]

http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/towler_pauli.pdf

By comparison, I am saying that the deBB theory allows one to *derive* these axioms from the deBB particle dynamics.

 

[GeorgCantor]

Why do you assume that it doesn't?

Does the wavefunction of the deBB live on configuration space or in real space? The 'imaginary' and 'real' parts of the wavefunction you mentioned earlier in the thread would in this case be the ones that tunneled through a classically forbidden barrier and the ones that didn't, right?

 

[Cthugha]

I am not presenting it as the 'holy grail' of QM. That's your projection. I have simply pointed out that deBB provides the quantum force explanation of the PEP which Zarqon seeks. And this is something which you cannot get out of the textbook accounts of the PEP.

Funny, I always considered the possibility to express the PEP in terms of a quantum force in deBB as an argument against deBB. Common QM treats bosons and fermions on equal footing. Probability amplitudes leading to indistinguishable bosons/fermions ending up in the same state interfere constructively/destructively. The difference between PEP and - for example - photon bunching is just one sign in the interference term.

In common QM you see the tendency for photons to bunch also getting stronger with the factorial of particles involved. Accordingly also the number of (in principle possible) states not available due to destructive interference increases in the same manner, thus explaining the "quantum force" as a change in the ground state of n indistinguishable fermionic particles as opposed to n independent particles.

In the first round of deBB, photons were not supposed to be particles, so this close analogy does not arise there. Even though more modern approaches to deBB also give rise to the possibility of treating photons as particles, it is still not possible to treat the PEP and its sister effect photon bunching on a similar ground: While attributing the PEP to a quantum force works, assuming a force pushing massless particles around does not make too much sense. You need to treat these cases on unequal footing.

 

[nismaratwork]

How did this turn into a dBB for sale thread?! dBB seems to just keep up with QM, and its best feature is not being annihilated by Bell, and little else. The whole notion of a pilot wave raises issues such as the one GeorgCantor asks in #47. If anything here is ad hoc, it's dBB with pilot waves guiding Schrodinger trajectories.

 

[Count Iblis]

The pilot wave guides PF posters to talk about dBB on QM threads.