In mathematics, principles play the role of an offer one cannot reject. The have no proofs, otherwise we would call them theorems. Nor they are axioms, otherwise one could reject on of them and accept another. In physics, I think, they should play the same role.

In my view, in physics 'principles' play the role of 'axioms' in mathematics. From this perspective, the question can always be raised as to whether a given set of 'principles' is the minimal set, that is whether all elements of the set of 'principles' are mutually independent. Physics being an empirical science (the mathematical language used in the exercise of in particular theoretical physics should not distract from this essential fact), the elements of the set of 'principles' are subject to continual change and variation, depending on the latest experimental observations and the attendant theoretical developments.

Incidentally, not all items in your list of 'principles' are principles. Thus while in this list 'the principle of least action' is a genuine principle, 'the uncertainty principle' is not, it being rooted in the algebraic structure of quantum mechanics, which is based on non-commuting operators; it was a 'principle' according to Old Quantum Mechanics (that of Bohr and what came later, until around 1925), but no longer so since the completion of what we call Quantum Mechanics (more completely, Modern Quantum Mechanics). Similarly as regards 'the exclusion principle', with the proviso that it is a 'principle' in the framework of non-relativistic quantum mechanics, but not in that of the relativistic quantum mechanics where spin and statistics are tightly interwoven.

I tend to think of physics as a set of an immutable space of descriptors and their eom. It turns out that some principles like the minimal action principle (which presupposes a lagrangian exist) work so often that it is a good starting point.  Quantum muddles my desired state of things by including a measurement axiom and a vagueness in the notion of field and particle but I do believe that a purely wave based theory will triumph. Stat mech requires principles based on partitions which I also hope will be replaced with dynamical explanations.  The second law has an almost independent status but should follow from a quantum dynamical theory if it is universally true.  Physics is in a fractured state right now so we need principles to start some calculations... for now.

Dear Farid, what would you tell about the energy conservation law? We obtain it only as one of consequences of the Noether theorem, but turn it into a principle which is valid for all physical systems. Well, on one hand, this theorem is valid only for systems which obey the action principle and on another hand the principle is incompatible with retardation. For example, if you take retardation into account when considering the Earth-Moon system, you find non-zero torque which breaks both energy and angular momentum conservation laws. The Noether theorem does not work in this case. It can be applied due to lack of Lagrangian for a system with retardation. As a result -- no conservation laws and energy conservation is not a principle.

@Zafar, this sounds like the problems one runs into with action at a distance principles cf. Feynman and Wheeler.  If you include the fields in the lagrangian, conservation is restored.  It should be noted that these only follow in the case that rotational and time translational symmetry exist.  For GR this is often not the case.  

Dear Clifford, it looks like a strong belief, but still nobody knows what is energy of gravitational field. Do you suggest to define it such that the total energy of the system is constant? On non-relativistic level I cannot do this but I can see why radius of the lunar orbit grows.

There is a quasi-local definition due to York/Brown.  In GR one often does not have such a symmetry so there is not expected to be a conservation law.

Anyway, I liked our discussion, thank you and Rajat for stsrting it.

the systems theoric order is:

..., law, principle, theorem, .....

Principles are heuristics, I believe.  They guide us to understanding of the Why of some situation.  Or raher, they guide us to hypotheses.  It still remains, in a new situation, to determine the What (laws) and the How (laws again, mechanisms) and the How Much (quantify).  So, for instance -- energy discrepancy --> hypothesize new variety of energy.  One must not elevate principles to laws -- that restricts their ability to encourage hypothesis formation.

Very good discussion indeed. Thanks a lot to all participants.

By the way, dear Hessami, Could you please give the complete hierarchy as suggested partially by you ?

Regards,

Rajat

Dear Rajat

The complete system theoric hierarchy is as follows:

World View, Paradigm, Theory, Law, Principle, Theorem, Axiom, Hypothesis

Thank you Dear AGH for the prompt response. Would you please define each for the sake of clarity of understanding?

Thanks and regards,

Rajat

>    The complete system theoretic hierarchy is as follows...

Says who?

And where is my favorite : a plain old hunch? (just me...)

Dear Rajat

Apologies for not being able to elaborate on these in a short space like this. The best way to make sense of the quoted hierarchy elements is to consider their affinity with objective truth with the World View being the most scientifically undisputed category of knowledge/insight and the hypothesis as the weakest hence the start of a postulate.

good luck

Dear Benham Farid,

The fact that spin-statistics theorem has been proved, does not any way lessen the status of Pauli principle as a fundamental principle even in QFT. In fact the anticommutativity of fermionic field operators is just another way of implementing the same old pauli principle only under a new title. So it remains a fundamental principle.

same thing goes with uncertainty principle as well. Non-commutativity of incompatible observables is just the same old the uncertainty principle restated.

Hope you appreciate this point.

Regards,

Rajat

Dear Prof. Alexander E Kaplan,

Thanks a lot for joining this discussion.

Could you please elaborate on what you think of principles vis-a-vis laws? We may disagree but let's see what comes out in the end.

Regards,

rajat

Dear Rajat, I strongly disagree with you. Regarding the Pauli principle, consider a Hamiltonian describing N identical particles. The particles being identical, this implies that the Hamiltonian commutes with the elements of the N-permutation group, so that eigenstates of the Hamiltonian can be classified according to their permutation properties (this on the basis of the mathematical fact that two commuting operators can be simultaneously diagonalised). By definition, one of the subclasses of eigenstates corresponds to fermions and the other to bosons (I shall not go into discussing anyons, and suffice to mention that my present argument can be extended to include them as well). If we now link the relationship between spin and statistics on the basis of the first principles (via the spin-statistics theorem), then there is no longer any place for the Pauli Principle as an independent principle of physics. Note that an eigenstate that changes sign under an odd permutation of its arguments is necessarily vanishing for any of its two arguments coinciding.

By your own statement, my above irrefutable argument proves the correctness of my previous statement with regard to the uncertainty 'principle'. 

Thank you Benham Farid for your explanation.

By the way, perfect identicality should mean that the results for all permutations should be the same wave function and there should be no phase factor coming in each permutation. The old formulation of quantum indistinguishability was accepted earlier since it was thought that phase factors in the wave function are unobservable. But now the Aharonov-Bohm and also Pancharatnam-Berry phases of the wave function are observed experimentally, thus making the earlier arguments look rather old-fashioned.

Thus In case of perfect indistinguishability the first as well as the nth permutation of any pair whatsoever should yield the same wave function. And therefore we should get only bosons and Pauli principle becomes a fundamental new principle necessary for fermions.

Regards,

Rajat

You are welcome Rajat.

I continue to disagree with you. What I wrote regarding the N-permutation group of a system of N identical particles, can be found in Chapter IX (Identity of Particles) of the book Quantum Mechanics by Landau and Lifshitz (Vol. 3), where for mathematical details the interested is referred to the Group Theory books by Weyl, by Hamermesh and by Kaplan (see also the book by Wigner -- in these books, see the sections dealing with the Symmetric Group Sn and its representations). Regarding permutations, distinction must be made between even and odd permutations, which form two disjoint classes of the elements of Sn. It is the odd permutations under which fermionic/bosonic states distinguish themselves from bosonic/fermionic states. Note that by calling those identical particles whose N-particle states change sign under all odd permutations of their coordinates fermions, and those that do not bosons, there is no room left for a 'principle' called Pauli principle: identification of any two coordinates of a fermionic state (anti-symmetric under the exchange of any pair of its coordinates) automatically results in the trivial zero state.

Regarding phase, the old idea about the arbitrariness of the phase still holds (this is reflected in the principle of gauge invariance). The Berry phase is a geometric phase, brought about by parametrically adiabatically changing the Hamiltonian of the system; if the space in which the Hamiltonian is parametrically changed is not simply-connected and the closed trajectory of the Hamiltonian is such that it cannot be shrunk to zero without leaving the space, then one will have a non-trivial geometric phase. The Aharonov-Bohm effect to which you refer corresponds to one such case, in which the vector potential is singular, rendering the relevant space a non-simply-connected one.

Thanks a lot Dear Benham Farid for your kind reply.

I still fail to understand how the idea of identicality or indistinguishability of particles can go together with sign change for fermions unless you add it from outside through Pauli principle. It is a matter of definition as you say, yes. Because by such definition we are just making a restatement of Pauli principle in terms of permutation symmetry which also includes bosons (the commuting ones ) in the scheme.

Indistinguishability should by definition mean "no change under permutation" because the particles are indistinguishable, you know. Thus, even a sign change for fermions cannot be admitted for indistinguishable fermions, strictly speaking. Why we admit it has to do with undetectability of a sign change or in general a phase in the wave function.

No principle in QM would be violated if a phase became observable in future like it has happened foe AB and Pancharatnam-Berry phases. If one goes by traditional QM arguments, even these phases must have been unobservable.

Regards,

Rajat

You are welcome Rajat.

Regarding the distninguishability / indistinguishability of particles, we put that in by hand when setting up the Hamiltonian of the system. To avoid the time-consuming task of writing mathematical expressions on this page, for my arguments I refer you to page 676 of the book Solid State Physics, by Ashcroft and Mermin (I use this book simply because at this very moment it is on my desk). In equation (32.5) of this book one has a non-relativistic Hamiltonian for two particles, treated quantum mechanically, that in isolation would belong to atoms 1 and 2 centred at the fixed points R1 and R2. It is assumed that the quantum-mechanical particles (which are actually electrons, and thus fermions -- but I refer to as particles so as not to fix their statistical nature at this point) are negatively charged, with charge -e, and the points R1 and R2 are positively charged, with charge +e, so that in isolation each particle is bound to its specific classical point charge centred at Ri, i=1,2 (this is so in the case at hand where the interaction between the charged particles is the long-range Coulomb potential). The two particles are indistinguishable by the fact that the Hamiltonians h1 and h2 describing them in isolation are identical: they contain exactly the same mass m and charge -e (see Eq, (32.6)). As a result, the total Hamiltonian of the two-particle system H = h1 + h2 is invariant under the 2-permutation, or it commutes with the symmetry group S2 (see my previous comment on this page) -- this you can directly verify. S2 has two elements: the identity operation E, and the exchange operation O. One has E (1,2) = (1,2) and O (1,2) = (2,1). With Ψ(r1,r2) denoting an arbitrary two-particle eigenstate of H (this eigenstate parametrically depends on R1 and R2), one thus has E Ψ(r1,r2)  = Ψ(r1,r2)  and O Ψ(r1,r2)  = Ψ(r2,r1). Note that one would generally denote such arbitrary eigenstate by Ψ(x1,x2), where xi = (ri,σi), with σi denoting the spin index of the i-th particle; the group elements {E,O} operate on xi. For convenience, I neglect spin here. The two-particle eigenstates of H that change sign under O describe fermions and those that do not change sign under O describe bosons. There is no third possibility, an aspect that remains in force for arbitrary N (in describing a system consisting of N identical particles) when subdividing the elements of SN into odd- and even-parity permutations.

Consider now the variational wave functions in Eqs (32.8) and (32.9) of the book by Ashcroft and Mermin (the interaction potential being the Coulomb potential, the problem at hand is not exactly solvable -- this is not the case for arbitrary two-particle problems; for instance, for a parabolic interaction potential the two-particle problem can be rigorously separated into two one-particle problems). In the book by Ashcroft and Mermin the particles are assumed to be electrons, and thus fermions. The variational wavefunction in Eq. (32.8) is orbitally symmetric, and is anti-symmetric under the exchange of the spin coordinates (not shown) and that in Eq. (32.9) is orbitally anti-symmetric, but is symmetric under the exchange of the spin coordinates (not shown) --- these facts with regard to the spin components of the wave functions are reflected in the names 'singlet' and 'triplet'. If you now consider the orbitally anti-symmetric variational wave function in Eq. (32.9), you will notice two different single-particle states ψ0 and ψ1. If you would replace ψ1 with ψ0, or ψ0 with ψ1, you would get the trivial zero state. What is traditionally called 'the Pauli principle', namely that two electrons cannot be put in the same state, is just explicit in Eq. (32.9). With regard to the variational wave function in Eq. (32.8), putting the electrons in the same spin wave function (not shown -- often the spin wave function associated with particle i is denoted by χ(i)), one will similarly obtain the trivial zero state.

To summarise, first, indistinguishability of particles is related to the Hamiltonian of each particle in isolation; when two particles are distinguishable, their respective Hamiltonians in isolation differ from one another. Second, in non-relativistic quantum mechanics, statistics of a particle (whether fermion or boson) is not reflected in the Hamiltonian associated with that particle in isolation; only when we consider an assembly of identical particles, this statistics manifests itself in the symmetry property of an eigenstate under the operations of the elements of the relevant permutation group. The strict subdivision of the N-particle eigenstates of a Hamiltonian describing N identical particles (see above) into two non-overlapping classes, one changing sign under the operation of an an odd-parity elements of SN, by definition describing fermions, renders any appeal to the Pauli 'principle' utterly redundant. I emphasise, once fermions are defined as particles whose N-particle states, with N >1, have the latter property, there is no room for the Pauli 'principle'. The latter was a genuine principle for so long as one observed that particular lines in the spectra of certain systems were missing, without knowing that the physics of these systems are governed by Hamiltonians with certain underlying symmetry, and the corresponding eignestates falling into the irreducible representations of the group associated with this symmetry. We are long past those times!

Incidentally, one can also have systems consisting of a mixture of particles that are distinguishable, are bosons and are fermions. The above arguments are easily generalisable to these systems. The only relevant issue is the way one defines the symmetry of the underlying N-body Hamiltonian, where N now can be expressed as N = n1 + n2 + ..., where ni denotes the number of the i-th particle species in the system, which may be either fermion or boson. Grouping the relevant coordinates in an appropriate fashion, one will have a symmetry group consisting of the direct product of the groups Sni.

Because I have already expanded too much on the issue of whether the Pauli principle is a genuine principle or not (it is not), I leave out any further discussions with regard to the phase. Suffice it to mention that my earlier remarks on the subject remain in force.