 we went through all these infinitesimal translation, then I entered through parameter space, then the generators associated with such a transformations and whatever is the number of parameters you will also have the number of generators ok. I am just warming you up before we get on to ok. And then I said that rotations also in the last lecture we saw how the generator is angular momentum orbital angular momentum. And then the rotation angles are the parameters and you have 3 independent rotation angles. And then we then I tried to explain what happens in space time, space time you can have 2 inertial frames where one is moving with respect to the other with constant velocity or you can have rotations of the frame itself. So, Lorentz transformation will involve rotations in space and also the boost transformation which involves one space and one time, am I right? Everybody is synced in rhythm with whatever I have been talking so far ok. So, there will be 6 parameters and 6 generators basically whenever you have a transformation you do see that the number of generators is equal to the number of parameters ok. And then I said that what is the symmetry of a system you can look at what is the symmetry of the system. Harmonic oscillator is not translation invariant, but your free particle has both translation symmetry and rotational symmetry. Then the corresponding generators of those transformation should commute with the Hamiltonian is a condition for looking at what is the group symmetry possessed by the system ok. So, that I left it and I we did I did tell you to check whether Hamiltonian p squared over 2 m commutes with L x L y L z and check it out whether it is 0 or not I do not know how many of you have done it, but please check it out. And then we went through this explicit infinite seminal rotation how a position vector the bold phase refers to position vector notation even though I am not putting a vector here this is the position vector and it is a cross product of 2 vectors ok. And then using the usual principle of how the wave function is going to remain the functional form of psi will change to psi prime r will also change to r prime and using this you can find what is the transformation which relates your wave function under rotation ok. So, this also we went through and for an infinite seminal rotation besides deviation from identity is the orbital angular momentum I have suppressed the h cross, but remember that could also be a h cross ok. And then I said that you can try to find that the order of infinite seminal rotation most of the books says infinite seminal rotation is commuting which is true as long as you keep up to order delta theta, but if you keep up to order delta theta squared then you see a deviation and if you try to write explicitly using the first expression identity minus parameter delta theta which is having a direction direction n hat unit vector dotted with the orbital angular momentum vector. So, if you put in this explicit expression into this then you will see that the difference at order delta theta squared is how many of you have checked this did any of you have checked it is correct and that automatically gives you the conventional algebra for angular momentum orbital angular momentum the commutator between L x L y turns out to be I L z ok. This is familiar from various directions, but for the purpose of this course whatever I am doing here I could have done for any transformation and then I should end up trying to get a algebra for the generators. By algebra I mean the commutator brackets amongst the generators must always give you a linear combinations of the generators ok. So, that is what we mean by an algebra commutator bracket of the generators any two generators because here you have only three generators commutator bracket of any two generator will give you the third generator here, but in general it could also be 0 if you had momentum linear transformation translation which was momentum p x p y commutator was 0 ok. So, it could be either 0 or it could be a linear combinations of the generators which are in that set which constitutes a transformation is this clear ok. And then we went on to our familiar orthogonal group of rotations in three dimensional space and then I said that there can be proper rotations or improper rotations improper rotations will have a determinant to be minus 1 and O 3 includes proper and improper rotations. So, it can have determinant plus 1 and determinant minus 1 and someone came and asked me what is this direct product? This C s is just a set which involves identity and an inversion matrix ok. So, this is a discrete group the C s is a discrete group and it is not a continuous group. Why it is not a continuous group? To go from identity to minus identity you cannot continuously go. If you have to continuously go you should be able to write that minus identity as identity plus infinitesimal transformation times generator you cannot do that for an inversion. Inversion is a discrete transformation you have a vector like this it is not continuously going to this place it is you know it makes an abrupt transformation. So, you cannot achieve such an abrupt inversion by infinitesimal steps the way we did for translation the way we did for rotation is a finite rotation can be seen as n steps of infinitesimal transformation that cannot be done for inversion or parity time reversal those are all discrete groups the way we saw in the first half of the semester ok. But what you can show is that the O 3 is a direct product of SO 3 times C s C s has two elements. So, you will have for every element it will give you two elements one with determinant plus one the other one will determinant minus. Is that clear? I also specified that SO 3. So, you can also look at why there are three parameters by writing R which is a 3 cross 3 matrix right and you have this condition that R R transpose 3 cross 3 is identity. So, R is a 3 cross 3 matrix with all the entries being real right. So, you have this to be this has nine elements to start with right and this condition. So, this is a condition or constraint you can call how many conditions are there. So, you will have diagonal element should be one of diagonal elements of this is 0 ok. So, you will see that how many conditions you will get out of this there will be three diagonal equations and three of diagonal equations. So, the number of conditions is 6 ok. So, in general any 3 cross 3 matrix will require 9 real if it is real entries will require 9 real parameters to define a 3 cross 3 matrix, but because they represent rotations what is rotation doing? Rotation takes care that x square plus y square plus z square which is R vector dot R vector remain same under rotations ok. So, this means it is R dot R transpose dotted with R R. So, this one is your R prime and this one is your R prime and this is the dot product. So, this is what we mean by saying it is constant or invariant under rotations ok. So, this is going to be invariant under rotations, this objects are going to be invariant under rotations. These are three dimensional real vector space R R can be treated like operators 3 cross 3 matrix representations of linear operators acting on the three dimensional vector space and these operators or linear operators have 9 real elements ok. It requires 9 real elements to specify your rotation matrix. However, they are not independent they have to satisfy this constraint which is 6 constraints ok. So, the number of independent elements turns out to be 9 minus 6 which is 3 and that is why you require 3 parameters study rotations. In three dimensional space if you do rotations in two dimensions can you say the same argument what will be the argument? There be 4 elements required for R matrix 2 cross 2 matrix and then how many constraints? 3 constraints. So, you will have only one which is rotation about an axis perpendicular to the plane clear. What happens if you go to 4 dimensions? 4 will be 4 cross 4 constraints in general can be written as n c 2 right. So, suppose I look at rotations in n dimensional space what do I mean by n dimensional space? You do not have only x y z coordinate you have a vector space which is not just x y z it is x 1, x 2 up to x n right. Any arbitrary position vector will be a i times x i summation over i running from 1 to n that is the meaning of n dimensional real vector space. Each of these coefficients are real coefficients and you can have linear operators acting on this vector space. So, you will have n squared elements minus constraint what is the number of constraint n c 2 what does that give us n c 2 or n into n plus 1 by 2 n into because the diagonal element should also be included. This is only of diagonal elements you should also include the diagonal because this condition diagonal equation plus the off diagonal equations off diagonal is n c 2, but you will also have a diagonal. So, finally, what do we get from here? I do not know check it out whether you get n into n minus 1 by 2 anybody can check it. So, what is that going to be for n is 2, 2 c 2 is 1, n is 3, 3 c 2 is 3 for n equal to 4 how much it is? 4 c 2 is 6 and so on ok. So, these are the n c 2 is the number of parameters which is also equal to number of generators. So, the r which is n cross n is an element of is an element of S o n such that r vector which is this dotted with r vector should be r prime vector dotted with r prime vector where r prime is this one is r n cross n on the r vector and this one is r transpose you understand what I am saying right. If you remember this three dimensions extrapolating it to n dimensional vector space real vector space is straight forward. So, you can ask why am I doing this? So, many times it happens that these symmetries are seen n systems. So, we want to always exploit such symmetries and get more physics out of the system like hydrogen atom there is a symmetry which is called Rangel Lenz vector. Classical mechanics if you have done you would have seen that there is a quantity called Rangel Lenz vector which is conserved. Just like Hamiltonian sorry just like Hamiltonian commutes with L's components it also commutes with the Rangel Lenz vector. So, you want to combine them and make it look like some kind of an abstract symmetry. So, you will see at some point that I can show a hydrogen atom Hamiltonian will commute with L's it will also commute with Rangel Lenz vectors, ok. But what you can show is that the Hamiltonian with L plus or minus a, ok. This will turn out to be satisfying an algebra which we will call it as SO 4, ok. Small SO 4 is to denote the algebra and you will also see algebra is for the generators and if you exponentiate it as generators with parameters you will get the corresponding group. When I write the group I will put the S and O to be caps bigger letters, ok. So, that is the notation which is followed that this is smallest small o this is algebra, Lie algebra, ok. So, you can write the Lie algebra for them which have not done here, but at some point I will give it as assignment problems where you will do the algebra L x L y L z you know the algebra, but you do not know how A x A y A z algebra is you should also know the algebra between L and A, ok. So, then only it will become a closed algebra. A closed algebra is any commutative bracket between the generators, there are 6 generators SO 4 has 6 generators, there are 6 generators. I want to write the algebra for the 6th generator, clear. A is the Rangel Lenz vector, you know what is the Rangel Lenz vector that which keeps the power bit closed and it is proportional to p cross L minus some constant times R or something, ok. So, R hat. There are some interesting aspects of looking at is this a polar vector or an axial vector question. So, much doubt some of you are saying axial some of you are saying polar it cannot be both it is not taken as a height. L is an axial vector, p is a polar vector, polar vector with an axial vector is going to be under parity what happens? It changes sign this anyway changes sign this is a unit vector position vector. If this changes sign you know it is a polar vector, you cannot add a polar vector and an axial vector. L is an axial vector, but p cross L is a polar vector. So, this is going to be a polar vector, ok. So, this is for going from SO 3 to an abstract SO n group, ok. So, these will be the SO n group. Now, I also want to slightly tell you about space time, ok. So, far I looked at it as a three dimensional space and an abstract n dimensional real vector space, but I want to do a slightly variant that I want to look at space time. So, let me do the same thing for space time, yeah, where is 6 components? So, I argued with you that N c 2 decides for you the SO n group, number of parameters. This is also equal to number of generators, there are 6 generators, N c 2 will give 6 only for N equal to 4, SO N equal to 4 is what will give you 6 generators, is that clear? Number of generators is not the N of SO n. Number of generators decides for you how many independent elements you can have in the matrix, which are linear operators acting on that vector. And that will turn out to be with this constraint, it will turn out to be N square minus N, these are the constraints, diagonal elements and off diagonal elements and what you get is the number of independent parameters and that decides for you the number of generator, which we have seen by examples. We did Lorentz transformation also, we did rotations in 3 dimension, I am only extrapolating for general SO n group how to do it, is that clear? Now, I am going to do it for space time. So, space time means what? So, let me look at 3 plus 1 dimensional space time. So, what we do is we write x mu by x mu I mean c t x square sorry x y and z, this is the meaning of the 4 dimensional still a real vector space right and I will do a transformation, which I call it as lambda acts on x mu to give me an x prime mu and lambda what is the dimension degree of these matrices? 4 cross 4, 4 cross 4 matrices acting on x mu will give you a new space time vector. What is the property required? You require x squared plus y squared plus z squared minus c squared t squared to be same as x prime squared plus y prime squared plus z prime squared minus c squared t prime squared. c is the velocity of light and you require this. Under any transformation between two inertial frame, if this is preserved then you will get the you know the system has preserved the symmetric. That is why you have all the equations of motion having the same form f equal to m a will remain as f prime equal to m a prime using a subset of this which is rotations, but you can write all your Maxwell's equations in this way ok. So, this is the requirement. So, in this notation which I have written here x mu as a column vector this way I can write this also as a dot product just like I wrote here I can do that there. So, what is the dot product going to be? You have to remember if x mu is this way we define small x mu to be c t minus x minus y minus z or equivalently you can also define it as eta mu nu x mu, where eta mu nu is a diagonal matrix. What is the diagonal matrix? Let me write it above. Diagonal matrix is 1 minus 1 minus some books follow the reverse, but there is an overall negative sign ok. So, just follow one notation ok. So, that will give you back your x mu. So, there are ways of writing this equation. How will you write it? Just like the way I wrote here I can write it as x mu x mu or equivalently I can write this as x mu eta mu nu x mu all or one and the same the repeated index has to be summed up ok. So, this explicit form is x square plus y square plus z square minus c square t square ok. So, I wrote the other way around here I think did I probably you can put this to be negative. See this is what I make a mistake ok. So, then you will get this to be the same thing. As I said overall negative sign can be taken out and what do I what am I saying? If you do this transformation lambda which is a 4 cross 4 matrix it takes x mu to x prime mu, but you can say that this thing should be same as the corresponding dot product is defined using eta mu nu eta mu nu does not change ok. So, what does that give me? It gives me that lambda lambda eta mu nu. So, lambda transpose eta mu nu lambda let us write that. So, this gives you a condition that your lambda eta lambda should be eta. So, now, you have some earlier when I did rotations I did not introduce an eta, but if I had I introduced an eta what will that eta be? It will be just an identity matrix. Now, you see that the time coordinate is distinguished from space coordinate and that is very essential for cause and effect to follow cause should happen in space time ok. So, that is why that this sign difference is important. You cannot say in space time it is plus c squared t squared. You could have asked me why do not I put plus c squared t squared, but if you do that in one inertial frame if you had event one and event two where one was a cause and other one was an effect when you go to another inertial frame it should be that both the events are seen by both the observers. So, the cause and event should follow the cause in any inertial frame that forces this negative sign in your space time. If I put a plus then there is no distinction between SO 4 or the Lorentz transformation. Lorentz transformation has to respect the cause effect sequence and that is inbuilt in this negative sign and that is inbuilt in your eta matrix. There are some of them which is plus, some of them which is minus ok. So, that gives you a constraint that you have 4 cross 4 matrix. How many independent entries are there? How many are there? 16 ok. These are the constraints. Any change in the number of constraints just because I inserted an eta just that earlier all the diagonal elements were 1. Now, some diagonal elements are 1 other diagonal elements are minus 1. So, the number of constraint is going to remain the same. Number of constraints is not going to change just because I introduced an eta. That eta keeps track of 1 with a sign and negative sign and 3 with positive signs ok. This is the way I am taking it. So, the number of constraints again here is going to be how many? 10. 4 diagonal and 6 off diagonal that is 10. So, 16 minus 10 is going to be so there are 16 real elements minus 10 constraints which adds up to give you 6 parameters and we have already seen what are the parameters. I called it as phi x t, phi y t, phi z t which are the boost where 1 inertial frame moves with respect to the other inertial frame with velocity v x. How is velocity v x related to phi tan hyperbolic phi of x t is v x by c right. So, those are 3 parameters. The other 3 parameters are your conventional rotations about axis which we called it as phi x y in the x y plane, phi y z and phi x z. So, it is clearly fitting in that the 6 generators which we saw elaborately for Lorentz group last time. I did not go over all the algebra of writing the generators, finding the algebra which I left it to you. If you do it there are 6 parameters and 6 generators and which is a consequence of this is this clear?