 And there is actually another point here, which is that this quantum kinematics hypothesis that every observable will be expressible in terms of q and p, can fail. And it fails royally its spin, spin is quantum number which you do not have any spacetime representation and you do not have any classical analog. So, when you get to spin you have to distribute in hand from your pockets. ज masc enchמע Bag scorp हemon ज़ कुछे मििपः्त है � bright य्हने तुर थंद कौत अरने घम کرटा कूर्टा लंं के ळारे बईऊए लेक धौई bring जो कुब अपन। C इप यह वेँ। आद मोंतम टिम। सो यहसर लिए सो विआ, उटर तुर विआ, मिक्स as well. So, you can encounter when you go to quantum mechanics new systems where you cannot break it down like this to it may not have a classical limit. So, any observable that does not have a classical limit like spin may not have a canonical representation like this. The other corresponding point being that in enumeration the spin half you have to quantize by anticommutators instead of commutators which is a very big jump because it has again no classical analogue right. The fermionic systems have to be quantized using anticommutators to account for Pauli principle. So, there are divergences from this visible even in the very simpler examples of quantum field theory. But in modern times we are living in many many more complicated situations where in fact quantum field theory itself looks in doubt ok. So, like this conformal field theory is that our people are studying you may not have a Hamiltonian description at all. So, you may not have so called canonical variables, but the previous part the first part is correct. So, as you know this classical analogy was a simplification, but the modern things do not affect the fundamentals of quantum mechanics because there all you have to do is list all the possible commutators between all the possible operators. And if you have a recursive way of stating it then you are done you can state some ladder of ladder algebras between Virasura algebra and so on. So, if you can state that you may have infinite number of independent observables, but if you can state their commutation relations then you have fixed the quantum system. So, not all systems will have this luxury, but we try to make do with this for almost everything that was known that is known at present except in the strongly correlated condense matter systems. So, I am told if I am not mistaken that. So, superconductivity, superfluidity all of these were very exotic states they had to do with this bosonic enumeration of states and Bose condensation and also the fractional quantum hall effect which is which has a wave function which goes by the name of the person who got the Nobel Prize for it Laughlin. Laughlin wave function the wave function exists, but there is no Hamiltonian of which it is an Eigen state. They do not know Hamiltonian system for which it is in the list of its spectrum. So, we are at that borderline where this simply stated set of principles may need some extension, but there is nothing varied I mean it will be nice and exotic if you we find newer quantum systems than just this simpler ones, but there is nothing that threatens our civilization. Ok good. So, the second thing I can do this is the issue of quantizing weakly coupled systems. So, the next thing is and it is important to say that it works only for weakly coupled systems and if we talk about weakly coupled electrodynamics in scattering processes QED scattering QED as well as electro weak. All of this is weakly coupled electrons in a lattice speculously display this weakly coupled kind of phenomenon because you can think of them as free gas of fermions in the lattice, but that they are really quasi particles. So, both terms should be put in separate. So, you can continue to think as if there are electrons floating in the lattice, but actually it is a much more complicated situation to which in the situations where you are lucky you have that description available often called the Landau liquid Landau Fermi liquid. Think about this you have the lattice which is positively charged and you have electrons which are all negatively charged and the coulomb force is infinite range force because it goes only as potential goes as 1 over r. So, they are all actually interacting with each other every single electron is interacting with every single other one, but because somehow of this big positive charge that is given by the background and the way Fermi statistics forces them to stack up above each other they just flow around as if they do not see any interaction, but their interaction is then hidden in their effective mass. So, there is the right m star I think it is written like this which is d e by d k I think it is this with some h crosses to be included right. So, you define it like this and you know that the dispersion relation energy as a function of k has this band structure with gaps and when you reach here you have very large mass because the curvature becomes 0. So, as the curvature of d e by d k curve becomes 0 the mass becomes infinite. So, here the fermions do not move here they cannot move things like that, but these are actually quasi particles. So, what is beautiful is that every electron has a charge and a mass and this complicated system just looks as if it is a collection of individual entities with charge and mass except that the mass has to be redefined which is why we. So, these entities we call quasi particles mass has to be redefined the charge cannot be redefined because of gauge invariance you cannot have charge changing in the system the total charge may be renormalize once and for all, but it cannot be a k dependent difference because then the gauge invariance operation will fail ok. Gauge invariance requires a phase in the exponent and the phase has to be single valued. So, it can only go from 0 to 2 pi. So, all charges have to be compatible with integer multiplication. So, that saves the charge of the quasi particle to remain exactly as the electron charge, but the mass gets renormalized and then you can continue to think as if you have some kind of fluid of particles, but it is just an amazing miracle and that breaks down when you get to systems where you know condensation of various kinds occur ok. So, getting back this procedure of quantizing a quantum system correctly and as I said correctly means that you have to have postulate number 7 with you without that you are not doing a correct job. So, this thing was thought up by Falk and Dirac along the following lines. So, the first observation is that observe that the states many particle states have to be symmetric or anti symmetric. So, I will write the fermion part in the bracket that means that. So, suppose I had 5 part. So, suppose we have a list of quantum numbers. So, suppose I have single particle quant ok single particles states labeled as some set of observables alpha i then a general state. So, here you say alpha 1 i for number 1 then alpha 2 is for you know for number 2, but this is not correct because you have to symmetrize it. So, you really have to do a symmetry operation some over the permutations of this 1, 2, 3 n. So, the list alpha i specifies one particular list the list alpha j is another list ok. So, particle 1 could be carrying this and 2 can be carrying this, but you also have to take the case when 2 is here and 1 is here and so on. So, you have to sum over all the permutations. What this means is that thus really this is simply equal to n alpha alpha i and alpha j. So, I messed up by not say. So, think of the i and j as defining different sets of values right. In other words the states are labeled by how many particles there are in a particular quantum numbers set of quantum numbers. You cannot tell number 1 is with alpha i, but number 2 is with alpha j. All you can do is count. So, if alpha i recurs anywhere else all you have to do is say well there are 5 of these that carry this quantum number 7 of these that carry this quantum number. So, the states are labeled by this such that of course the n alpha i plus you know this adds up to n that is all you have ok. Now, if this is so, then we have a clearer opportunity and this is where our this psychological problem business is explained because we do have the number operator. So, when people are worried that I have 2 electrons and then they are coming apart and then I observe spin here then that spin gets determined. It has all to do with the fact that ultimately there is a quantum number associated with the number of fermions it is 2 that state has 2 and that can be observed because they do have a charge and the charge can be observed at a distance even without disturbing the system much. So, you know for sure that there are 2 in the case of photons we do not have a conserved number. So, there you have to be sure that you are not producing or annihilating a photon, but so I am saying this because there are lot of the entanglement experiments are done with photons, but I think they are able to control that part. So, the number has to make sense and the psychological conflict in this thing has to do with the fact that it is one state, but the number operator Eigen value in that state is 2 ok. So, they think there are 2 different entities. This cross product where we take one particle states and then string them together to make more is really a mathematical device ok. If I did not have this summation this has no physical meaning because if it is unsymmetrized then it has no meaning it is not one a physical state. So, the fact that we can fall back on single particle states to string together a many particle state has to do with this weak coupling. And the famous thing that I have left out of here is QCD. In quantum chromodynamics we have almost massless quarks and we have exactly massless gluons and there is no way you can count them individually. So, the and they are never weakly coupled they are weakly coupled only when they are being scattered at very high moment very high center of mass energy which means that their weakly coupledness lasts only for fleeting seconds fleeting moments. For most of the time they are blobs headrons which are all in a strongly coupled state and you cannot pull the quarks apart. So, quarks are prime example of where this construction will fail. So, the first step is that due to postulate number 7 all the states are labeled only by number number in each possible slot of Eigen values that are available. But if this is true we observe that such states are the complete list of such states is mutually orthogonal orthonormal we will say because if they are equal then of course it gives 1 and 2 is complete right. Because they are mutually orthogonal because if the number in any slot if I take expectation value of something else then number in any slot is different then it will give 0. So, you have to have exactly same numbers in every slot then of course you will get mod square of that state, but if any numbers differ then you immediately get 0. So, that is one property the second property is that of course every possible state of the system can be formed by linear combination of these. These and this is all there is if you list all the possible symmetrize states then any possible state can be written as a linear combination of these. So, this is a complete set. So, because of this fact that they are mutually orthonormal and complete you can think in reverse that therefore there must we are tempted to introduce hermitian operators of which they are Eigen sets right because they are complete and mutually orthonormal suggest that there are some hermitian operators of which these are Eigen vectors these are simultaneous Eigen vectors. So, it just occurs to me that how much damage you do to the quantum state in a observation process depends on the kind of up. So, there are things called weak observations. So, if you get into that language then there is lot of discussion there right. So, we can expect existence of corresponding operators which correspond to each number operator corresponding to each of these such that its Eigen value is the small n alpha. And one thing I have to add the two for completeness we will also need the vacuum state which is a no particle state. For a single particle there is nothing like a no particle state, but once you introduce these collective states you also need for completeness a state that is a no particle state. So, we also introduce set that there is all ns are 0 in it. So, then we can think of the spectrum to be fully complete with respect to all of these because you know because some of the you will find that many of the ns are 0 that say in some state, but then you have to consider the possibility that all the ns are 0 as well to make the whole spectrum of states complete. So, that is the ground state alpha in modern times we know that much mischief lies inside this so called vacuum state although we think it is no particle it is not strictly no particle because it keeps doing something it has quantum fluctuations in it. But anyway we are coming to the end of this construction and will stop in 5 minutes which is that now that you understood that there are number operators and which also satisfy they are all mutually commuting. So, now we therefore in reverse since we are so familiar with quantum mechanics we are tempted if I have number operator to introduce a and a dagger out of which I can get the numbers. So, you have to exactly same list to then you get one and otherwise everything is other things are 0. So, you can always introduce this and that will reproduce this algebra. So, this expression in terms of a a dagger is a great calculation device now. So, this gives you a new calculus if sometimes somebody ask me you know somebody was otherwise very accomplished engineer and mathematics person. What is new mathematics of quantum mechanics? Well the new mathematics of quantum mechanics is that the a a dagger as the elementary operation then allows you to calculate s matrix elements. So, you generate the scattering processes by using these and using their commutators with the hematonia. So, thus so finally, we observe for and of course, anti commutators for fermions and note that the observable also is symmetric in the and in terms of particles. So, the observables have to automatically be constructed out of a s and a daggers only. So, thus every operator. So, what does one need in life? So, every Hermitian unitary or projection operator needed can be constructed out of and therefore, every calculation that you need to do in this theory can be boiled down to computing commutators or anti commutators of a s and a daggers. So, for so of course, so far the system was free, but we said weekly coupled. So, now you can introduce some interaction. So, I will end by writing that a generic interaction. So, the free Hamiltonian interaction what should we write v alpha k l beta j m. So, I have some set alphas and some set of betas which are l in number and m in number and I introduce it just counts how many distraction and how many creation operators. So, you write that many here. So, you can write out some expression like this where l number of particles come in and n number of particles go out and we have to characterize the v correctly. So, that we get meaningful field theory. The v should effectively give a localized. So, v can localized a properties that the v has to basically ensure and what that means takes is the content of all quantum field theory. How to construct those is what the quantum field theory is about. We will stop here. So, incidentally if you want to read up this part this last this Fock Dirac construction is given in Merzbacher's textbook. Merzbacher's book with some variation in notation is the same physics the same things he talks about and very nicely moves on to the so called many body theory. But we will break from here onwards we will go on to the field notation. Of course, as you know fields can be recovered now by doing Fourier transform from the alpha space to a coordinate space. So, generically the alpha space is simply momenta k the normal way of labeling free particles is their momentum and spin. So, if you Fourier transform from momentum to coordinate space you get the space time fields. Most people most books almost all the books try to tell you that the field is some entity given by God and that we shall try to quantize it and they say oh look do not be afraid of a field because you have seen coupled oscillators and then. But this is actually the construction. So, after they do it they automatically get both and Fermi systems because they put commutators and anticommutators. But why you did it has to do with this underlying postulate 7. So, this is the real construction of a weekly coupled field theory. But to lot of people it seems that if you start directly with the field then you are doing something more advanced and maybe you have you are accessing and non-weekly coupled theory as well by making those postulates. But I do not think so.