 This is for weekly couple systems. In connection with the quantum basics, I want to add something comment on QM versus Cm, quantum mechanics versus classical mechanics. There is much debate and doubt and controversy about quantum mechanics being so bizarre and then what is really happening is it you know these words ontology and epistemology is it. Epistemology is how you organize knowledge whereas, ontology is what exists what is actually out there. So, how do you know whether this is just a organization of how what you can know versus whether there is a absolute reality out there. So, a lot of philosophical things is going on. So, I just want to point out that there are as many elephants in the living room in the classical theory that people do not seem to worry about anymore and one of them is calculus. So, in classical mechanics we have the idea of derivative right x dot equal to limit delta t going to 0 of and similarly other derivatives d f by d x right some limit of delta x going to 0 of whatever. Did anybody in real life actually take the limit to 0 did they well when they did they found quantum mechanics before they reach 0 they began to find quantum mechanics. The smallest scales they reached was atomic scale, nuclear scale, LHC has reached some astronomically smaller scale than that, but nobody has reached 0 right nobody has explored 0. So, this idea that this limit exists is a fiction from the point of view of physics because no limit has nobody has got a stopwatch which can be calibrated to go to exact 0 the best cesium clocks we have have frequencies in whatever. So, you can tune your frequencies correct to parts per 10 raise to 18 whatever you want it is never 0. So, this limit was never actually taken in practice, but people so, firmly believe in it that the thing that classical mechanics therefore, made the world predictable and then you could provide x and x dot at one instant of time and the Newton told you how to go the future. Did the how precisely did they know the past it is just that the precision of all their measurements was so, poor that it looked fine, but in reality classical mechanics is essentially founded on fictional ideas of what is the continuum. The ideas of what is continuum are having made very precise and refined, but whether those ideas actually define the continuum the physics continuum we do not know. In fact, what we do know is that if we try to explore at the really small scales we find quantum mechanics. So, better be open minded about what is out there and learn it rather than try to be pedantic about what was learnt from larger scales. So, I think this is something that most people seem to not get and our cognition has to be expanded to absorb new things rather than trying to force newer things to fit into the cognition, cognitional devices developed earlier. So, that is enough of a comment. Now, let us go to this particular formulation of path integral. Let us begin with some fixing of nomenclature and notation. We write Schrodinger picture states in this notation psi T s. So, we will put subscript s for Schrodinger and it is a time dependent state function. Now, we can write it out as being equal to sum of energy Eigen states C n e raise to minus i e n t over h cross times the n Eigen state ok. So, where n is energy Eigen states. So, now we have another notation these things written like this with some Eigen values written in them with no time dependence or anything are basis states. So, we use the same kind of angular bracket notation, but n are basis states. So, the set n is basis set. On the other hand Heisenberg picture we write psi h the state is not time dependent and the relation is psi s is equal to e raise to minus i h t acting on this psi at t equal to 0 which we can define to be equal to psi h. So, this is sort of for convenience if you like, but well the time independent state is the Heisenberg picture state ok. And for Heisenberg picture state we will have a simple statement it will simply be C n n and of course, one should then write the revolution for Heisenberg picture and that is given by right for operators we have right. So, one can write down that statement of e raise to minus i h t or etcetera and we are not going to need it fully, but this is just to fix the ideas that we write psi h we write psi s and then we will write like this. Now, this is the basic point I mean basic notation next we consider the kernel. So, quantum mechanics time evolution can be written out either as a. So, Schrodinger equation of motion sorry h t. Now, there is an alternative way of writing. So, instead of a differential equation we propose a t equal to t 1 to t to t and what we do is do we have a way of mapping directly from initial time to final time. Suppose I am not interested in detailed evolution of the wave function in any case knowing the wave function at in between points in reality would require it to measure, but you do not want to measure you are going you will collapse the wave function. So, suppose you only want to know it later. So, what you do is you introduce a two point function. So, we say like this psi x a k x t x 1 t 1 I am so sorry not like this. So, we have to go to wave function language right ok. So, this makes it the Schrodinger wave function no angularities you have the usual wave function. The definition of that wave function is we have taken the projection on this basis. Remember things that are unadorned by themselves are basis vectors. So, this is basis constructed out of x operators and so this is psi x of t and this we want to write out as b equal to. So, this is the idea. So, what we have proposed is a kernel. So, is the meaning clear? So, instead of solving the differential equation we want to propose an integral equation in which this will be found directly ok. So, that of course, requires lot of information because you have to know this two point function, but if you know it then you have the answer. Now, that can be determined as follows we say that integral d 3 x 1 sorry. So, we have to do only d 3 x 1 not t 1 to t very sorry right. So, this is not there it is d 3 x 1 only. So, integral d 3 x 1 of we introduce this time dependent basis what we are going to say is that we need to. So, the thing is to get the kernel try something like this try inserting a complete set of states right. So, this would add to 1. So, it is like taking this directly an overlap between the final and the initial. The point is that this needs a careful construction of this Dirac picture states. So, we need and the idea is that we make them instantaneous Eigen states of the Heisenberg picture operators by saying that we define. So, Rearad said that things that are nothing else in them except some Eigen values are basis, but now we are going to make the basis time dependent ok. And it is time dependent in the sense that it will return exactly the same Eigen value regardless of the time at which I measure it provided I use the Heisenberg picture time dependent operator. And I may as well write down now the. So, it is this is if you want Schrodinger picture operator. So, that is the relation between the Schrodinger picture and Heisenberg picture. So, the meaning of this is that action of X on this returns an instantaneous Eigen state and we can write this out as because this is time dependent. So, if we define by the opposite action e raise to plus i h t the unadorned basis then this will work. So, now these two will cancel X s acting on the unadorned basis will return the Eigen value X and then it will become. So, this is a bit of technicality and if you are reading this for the first time you may find that I am struggling over a technical point well it is technical unfortunately most books do not tell you that the Feynman kernel is a calculation between Dirac picture state and Dirac picture state. They just write in state and out state, but the in state and out state are actually this Dirac picture ok. So, now we go back here and then we can see what is the meaning of this kernel the kernel is actually. So, shall I just quickly repeat what we have done? We have introduced a time dependent basis suppose we ask for X t Dirac basis like this. The answer is that this Dirac basis is defined by this. The Dirac the basis that will satisfy this relation is defined by this ok. And now we will see the use of and so we just check that that is correct in the sense of you know this intervening operator and all does not affect anything we exactly regain this statement. And then our argument is that the kernel has to be defined in terms of the Dirac picture basis. So, note that so once I take the e raised to i h t out and see it as operating back on this it is basically. So, the wave function the usual wave function is either shredding a picture state projected on the usual basis or it is the Heisenberg picture state projected on to time dependent basis ok. And so now we can read off what this kernel is psi at a later time is equal to integral over the intermediate time d 3 x 1 of right because psi is Dirac picture basis projection of Heisenberg picture state. So, that is this and this and all we have done is inserted the identity in between. So, now we got the precise mathematical tool which passes muster as kernel. So, we can now say that therefore, the kernel is Q i t i Q. So, this part we can now write as so three lines this is equivalent to saying I have this then I have k of x t x 1 t 1 and now in the this psi again let go back to the original. So, then it will just look as if. So, psi the Dirac basis overlap with Heisenberg picture operator is same as ordinary basis overlap done shredding a picture which is just the wave function. So, it is psi which is the desired kernel. So, that is the precise meaning of the kernel provided both these are in the Dirac picture 0.1. So, end of you know preparatory remarks 1 now we go to the next part which is that Dirac's original motivation for the path integral and here again there is a danger of lapsing into very technical discussion. So, I will not do it because it is given in Dirac's book I can forward to you the notes and what we will do is that we will agree to some basic statement that the reasoning goes like this that in classical mechanics any two set of canonical variables can be related by a canonical transformation. By the way the word canonical actually comes Goldstein remarks from the word Kanun means law you know. So, kind of legal the formal the ok. So, by a canonical transformation and I hope that you have studied this transformations with small q and big q and you get from one to the other by differentiating the generate. So, there is usually a generator associated with it, but the dynamical variables at any time therefore, time evolution itself should right at any instant of time they are canonical they satisfy the canonical brackets the Poisson brackets. So, therefore, time evolution itself is a canonical transformation and comes the technicality what is the generator of this transformation and the answer is that it is the Hamilton's principle function which is defined in Goldstein's book as. So, this is a canonical you remember there are four kinds of canonical transformation all q's new p's all p's new q's or all q and new q. So, this is of the type all q to new q this is the generator. So, it is equal to the integral from x i t i to x f t f of and Hamilton's principle function is defined as p x dot minus h d t where the time integration is on the classical trajectory or one should say dynamic everything is classical mechanics. So, the integration. So, this is a appendage to the integral sign you might say what have you learnt because if you already know the dynamical trajectory then what is there to be solved, but this is a formal statement about what is the meaning of the canonical transformation what is its generator and what it consists of. So, in the language of canonical transformation the time evolution is a canonical transformation generated by this particular generator. So, that p final would be got as equal to d s by d q f and p initial well I mean there are vector signs, but so you put gradient if you like. So, p i equal to minus d s by because it is a lower limit this is. So, x I hope I did not write q I have notes in which I have used q and p here I started writing x and p. So, this is how the old and new this is how the generator works it is a function of the old momentum old coordinates and new coordinates and therefore, the new moment and old moment are derived from it by this. So, but you have to do the integration. So, do not mistake this for the action principle or the action. The action is a functional this is not a functional it is a two point function of real numbers real arguments which are only the end points and the p x h everything is function of time said that it is lying on the dynamical trajectory and you have done the integral along the dynamical trajectory, but from the two between the two desired end points. Then the object that you assemble of course, you would know it parametrically as a function of the end points x f t f q x i t i and that multivariate calculus function not a functional that function is called Hamilton's principle function. So, all this was formalized by Jacobi and I think the word canonical also goes to Jacobi. So, this theory of transformations. So, coming back to Dirac's motivation he says look for every transformation of basis that you can do in classical mechanics there exists a unitary operator which implements it in quantum mechanics ok. So, next recall postulate number which no third only third. So, we just said linear operators second one was observables or Hermitian operators and third was transformations are implemented by unitary operators. So, corresponding to a transformation in classical mechanics there exists a unitary operator in q m which will implement the same thing. And the operating q m u would be exponential of i times some generator g where g would be a Hermitian generator right. The unitary operator would be given by exponential of a i times a Hermitian operator which we usually refer to as a generator. Here we found that for the classical system the generator is this. So, if we exponentiate that we should get a unitary operator that implements this, but there is one major problem the it is a dynamical evolution and these quantities at different times do not commute. So, we cannot just exponentiate the whole integral what we can be sure of is that in the infinitesimal limit it would work.