 So, what is group theory? So, the main thing is symmetries ok. So, symmetries of pattern which you see in nature ok that plays a crucial role in trying to get some information about a complex system ok. So, you can you do not need to try and solve complicated equations to find the solution. Like for example, if you take hydrogen atom I am sure everybody knows hydrogen atom right. Hydrogen atom you all know that the energy is 13.6 E v divided by n square right. So, the energy in the hydrogen atom so, they are quantized and how do you find these solutions for E n you solve the Schrodinger equation. If you solve the Schrodinger equation it is a second order differential equations and that gives you minus 13.6 E v divided by n square. So, you have to solve that equation and what is the symmetry available in that system? It has a spherical symmetry or the potential which you write for the hydrogen atom is going to be invariant under rotation. So, the potential energy which we write is only dependent on the radial distance and the radial distance what is this one? R is the radial distance radial distance between electron and proton looking at hydrogen atom ok. You can take electron and proton to be like point like or you take center of the nucleus to the electron and if you do a rotation operation on an R vector what happens? R changes to a new R prime vector, but magnitude of R which is R radial distance that is going to be same under rotation. Rotation does not alter your radial distance. It takes the same radial distance, but theta and phi will change in the spherical polar coordinates and R goes to R prime under rotations. This is a rotation. I am just giving you a flavor of what group theory can do. As a conventional quantum mechanics problem you apply it in the Schrodinger equation putting this potential energy try and solve it right. So, you solve it how you know how difficult or how complex it is. It is still mathematically solvable sometimes if you have a complicated potential energy you may not be even able to solve it. Solving means finding the energy of that stationary states of the hydrogen atom ok, but what we could do is exploit such symmetries you exploit such symmetries and then you get the answer without even solving the Schrodinger equation ok. So, this is the sometimes powerful elegant way of doing complex situations by exploiting symmetries of the system. So, this is what is the theme of group theory. It is a tool you know if you want to break open this table you may need a good tool which is a screwdriver then you can easily remove it part by part. Of course, I can come crudely and start breaking with whatever I have then that is where you know you rigorously try and use all complicated mathematical equations. So, these are two different ways of looking at things and that is why group theory has become very important and this course is just to make you people appreciate such a tool and there are limitations. You cannot go more and more saying that I will not learn the other methods. I will also tell you till what extent this tool will be useful ok. So, sometimes suppose I ask you whether a particle in the ground state suppose let us take harmonic oscillate I am sure you have all done harmonic oscillation 1 D harmonic. So, 1 D harmonic oscillator what is the symmetry potential is. So, this has what symmetry x goes to minus x v of x equal to v of minus x. In this harmonic oscillator if you stall the Schrodinger equation find the wave function you can try and show that the expectation value of x how do you show that by using symmetries under this operation. So, let me call this operation as some g operator under this operation my physics which is contained in all the expectation value should not change ok, but under this operation I know wherever x is I have to replace it by minus x. This expectation value should be same as this expectation value. So, what does it mean I can pull out this minus x out because if you remember the wave functions when you write if there is some sign here you can pull out the sign out of that expectation value and you can show that expectation value is minus of expectation value which means expectation value of x is 0. Here I did not do any calculation I just exploited the symmetry that this is the symmetry of this potential energy for the harmonic oscillator. In fact, you can even say for what all powers you can say all odd powers you can say that it will be 0. Even powers can you say anything even powers you can say it is non-zero, but actual non-zero value you cannot predict. If you want to find the non-zero actual value you have to know the explicit wave function otherwise. So, it is like a process which will tell you that something is 0 something is non-zero a process of jumping from the ground state to the first excited state due to some dipole moment interaction is allowed or forbidden ok. So, suppose I have a ground state I have a first excited state a ground state particle in the ground state which is undergoing which is subjected to this potential. Suppose I make it interact due to a dipole moment interaction which is like this x operator dipole moment is q times x such an interaction can you get from a ground state and x operator to the first excited state is this going to be 0 or non-zero ok. Such questions can be 0 or non-zero can be predicted you cannot find the non-zero values ok. So, this is what is called as a selection rules. What is selection rules you will say that to undergo this transition delta m should be plus or minus 1 some selection rules you would have seen. Those selection rules are consequences that it is 0 or non-zero allowed or forbidden process a process of transition to go from here to here triggered by an external interaction is it allowed or not allowed ok. So, that much you can say it is allowed not allowed means a value will be 0, but allowed what value you will not be able to predict, but you can say it is not allowed clear. So, these are the things where the power of group theory is really very powerful and I want you to appreciate this for that you may have to learn a little bit of those tools I will take you through the path of what all is happening ok. So, this is the theme of this course. So, let me just go on to tell you what is my plan for this course. Group theory by Hammermesh is one book which talks extensively about discrete groups then the other book which I have been very impressed is one by Georgie even though it is written for particle physics people I will try to give you a you know the set of salient features how continuous groups have to be viewed. So, this will be one kind of a you know textbook which you could keep it at the background while you go through those lectures. So, in the process of teaching this course for the last you know I have taught this two three times in the institute with one of my former student Varun Dubai I have been trying to get this book to come in print form you can also you know take a look at this and I am trying to follow those writings which I have put it in this book in this lectures ok. So, the syllabus and plan is discrete groups in the first half of the semester. So, essentially I will try to focus on cyclic group, permutation groups, point groups then with matrices which you all play around how to focus on irreducible representations, grade orthogonality theorem, character tables and some applications in solid state physics. So, this will be the theme for the first half of the semester and there would not be rigorous proves like a mathematician we will only try to give a statement and supplemented with the examples and you know exercises. So, grade orthogonality theorem will be only said as a statement we will not prove it ok. So, proofs are not part of this course, the part of this course is to show how the tools can be applied to solving problems in physics ok, the group theory tools is that fine. The second half will be continuous groups. So, the simple examples like space translation, time translation, rotational symmetries which I have already indicated to you for hydrogen atom problem will lead us to slowly get the formal aspects of what the continuous groups are sometimes called as Lie groups and the corresponding algebras which are the Lie algebra. So, I will do focus on SU 2, SU 2 is not very different from the angular momentum algebra which you have studied and rotation groups which you have studied and that sets the path to go to SU 3 ok. Once we understand SU 2 clearly you can follow how to go to SU 3 and higher groups, but we will focus on SU 3 and in special theory of relativity you have all done Lorentz transformations. So, as more applications we will also look at the group which is called as a Lorentz group and its application ok. In one of these applications I will try and explain how to get this E n as 1 over n squared without solving Schrodinger equation. So, these are the motivations and aing which I have. Of course, it is every lecture is connected to the previous lecture and please do the assignments sometimes many of the problems will be for your own hands-on experience and that will help you to grasp things in a neat fashion otherwise you know you may not be you may be completely not in sync with the lectures ok. So, I just said discrete symmetry and continuous symmetry as a simple example I am showing that there is a square which has a. So, you have a square. So, the square you all know all of you know you cannot do 35 degree rotation and make it look like as if it was like the original configuration. The original configuration has 4 vertices and when you do 35 degree rotation the vertices points were all will all be distorted. So, the vertices will go into each other only if you do a 90 degree rotation right. So, it is a discrete rotation. So, the symmetry of a square is 90 degrees. Suppose, the square I put in some disturbance and it becomes a rectangle then what will happen? Will the symmetry be still 90 degrees? 180 degrees. So, these are things which you can start thinking you know any complicated polygon is given to you what is the kind of symmetries you have ok. So, basic rotation fundamental rotation has to be 90 degrees. Of course, you can do 180 degrees you can do 270 degrees and finally, if you do 360 degrees it is like an identity operation ok. So, for a square you have rotation by 90 degrees, rotation by 180 degrees, rotation by 272 and you will also have an identity operation which you can call it as rotation by 360. So, these are the 4 distinct operations. As operations identity is do nothing, then rotation by 90 will change the vertices the way it is shown here for clockwise and you can do 180 then the vertices will further shift as shown here right. You see that D has gone from here to here, A has gone here, B has gone here, D has gone ok. And if you do one more you will get a distinct state, but it will look like an originality that is why it is called as a symmetry. So, these 4 elements together this forms a set. So, you have a set set of 4 elements and you can introduce an operation which is the rotation operation as an operation. So, r pi by 2 if you do it twice 2 rotations by thing will give you rotation by 180 is that right. Similarly, you can do r pi by 2 with r pi you can get r 3 pi by 2. So, what have I shown a physical rotation by 90 degree is a group operation and if you do it amongst the set you can do it amongst the set. These rotations are going to give me an element which belongs to the same set clear. So, you have a set of elements and then you give you a group operation and the operation when we do it twice you know you can take one element from here another element from here you just multiply those 2 rotations ordinary multiplication you get back an element which is still belonging to the set ok. So, that is one this is what we call it as a closure property. So, group operation is just I would say that I will put a group operation which is just ordinary rotations which I am doing here ok. So, it could be sometimes if I write matrices suppose I write matrices that also I am sure you all know how do you write the matrices. If I want to find out rotation by 90 degrees you can use matrices then the group operation will be matrix multiplicity it will be matrix multiplicity. So, if I use matrix matrices for the elements of the set group operation will be matrix multiplicity. So, the group operation depends on what exactly you are going to you are going to do in this case I am doing physically rotations. So, you just do rotation then after that again another rotation that is the group operation or the product rule here is like this, but you can also do it like writing the explicit matrices and do matrix multiplicity. So, this is another way of doing it, but both are going to be representing the symmetry of the square. So, this set satisfies the closure property that I have already showed you that if you multiply any two elements you will get that element which will be in the same set. If it exceeds 360 you will bring it back to the first domain between 0 to 360 correct. So, closure property is there what about associativity property ok. So, there is also a. So, let me write I think I have this rule here. So, what is the group? So, you start with a set of elements in this particular case of a square there were only 4 elements. You have to also define a group operation and that group operation is dependent on the situation. If you give the a a matrix kind of a representation by representation I mean it is like you know a dress which you associate or pi by 2 you can treat it like a rotation by 90 degrees on a physical square. You can also say that I want to do the rotation by giving a presentation which is a matrix presentation for the rotation by 90 degrees. These are two ways of looking at it there can be many other ways of looking at it. Depending on that your group operation here the group operation is just doing physically 90 degree followed by 90 here it will be matrix multiplication. So, the group operation depends on the situation ok. So, you have to have a set and you have to also give a group operation and then it has to satisfy four properties. You should have a closure property which I have already explained take any two elements according to that take any two elements with the group operation ok. You should get an element C which should belong to that set. So, set which I calling it as G it should belong to that set that is what is happening here right. If you take any two elements that combination will give you another element which also belongs to that set is that clear and then identity element is what I have already said do not do anything do not do any rotation rotation by 0 degrees is an identity element. Any of these elements if you multiply with an identity element what happens it is the same. So, there should be an existence of identity, identity element has to be there in that set. If you do not have an identity element then that set will not be satisfying the properties to qualify for a group. So, you should have an identity element then you should also satisfy associativity property and it should also have an inverse ok. So, associativity is what if you take A combine it with B and then combine it with C it should be same as someone could combine B and C and then in A this property should also be satisfied. So, you can check it out that it is satisfying that. Inverses every element if you combine with another element let us say B and if it gives you identity. So, most of the group theory books the identity is written by the letter E ok. So, I am going to follow the letter E for identity element. You have to find at least for every element some element in the set A and B should be in the set such that this combination is identity. So, in this particular case for r pi by 2 what is B? Someone which will give me identity r 3 pi by 2. So, for every element you will find an inverse this element it is its own inverse it is also in the set ok it can also be self inverse. So, inverse is the this is associativity and then you have to find inverse. So, inverse for every element should be in that set if all the four properties are satisfied then you call that set to be a group otherwise it is not called as a group ok. So, I will give you few examples yeah I agree, but it has to be also complete you know for every element there should be an inverse in the set closure property will be trivially satisfied once you find that, but even if every element does not have an inverse the product should give you another element right that is always possible. You have to get another one which is saying that A B should be equal to E right. There is no necessity that A B has to be E condition is to be required for closure property. Converse is true if A B is equal to E it is inside the closure property, but just the closure property does not necessarily mean that every element should have an inverse. What is minus r pi by 2 mean? Yeah, anticlockwise will be r 3 pi by 2 is r of minus pi by 2 if you want to call it. So, it is not going to give you any new information you have to find a distinct element which gives a new information is that ok. So, I put it everything as clockwise going till 360, but you could also done let me go 180 degrees clockwise the other 180 in anticlockwise then you would have got r 3 pi by 2 as r minus pi by 2. Any other question? So, over on top of it if you have this requirement ok, if you have A B equal to B A ok the order if you do first the operation A and then a B or you do first B and then an A if these two are equal then that set with this four property satisfied which we call it as a group will be called as an abelian group ok. Abelian group satisfies this for all elements of the set not just some specific elements it should happen for any element of the group then you call it as an abelian. If A dot B even at least one element you find where this is not equal to this even if you find one element then you will say it is a non abelian is that clear. I started with a simple example of a square which has a discrete rotation which is 90 degrees and I found that set has four distinct elements for four distinct configurations and I tried to give you a formal way of looking at these expressions of closure, associativity and so on ok and every element should have an inverse. There should be an idg element which is like no rotation operation and we also have this property satisfied by the square. You all agree if I do rotation by 90 and then a 180 it is same as doing 180 and 90. So, it is going to be an abelian group that could be some cases where you will find it to be not an abelian group we will see all those things ok. So, just to give you some more examples take the simplest example set of all integers ok. So, let us take the set of so you are going to take a set which is all integers and have plus or minus right. So, this is the set. So, what is the idg element? Idg element is 0. So, the set has an identity element closure property any integers if you add it will give you a P which is also an element of the set then the third property is if you add m plus n, m 1 plus m 2 plus m 3 in a any way group operation is what I should have written the group operation. Group operation is addition set plus group operation ok. This is same as what is the inverse of every element m 1 inverse s minus m 1 which is also in the set and what more is it an abelian or not abelian a group it is abelian right. So, this is qualifies for it to call that set as a set of integers under group addition is a group is that right under addition group operation which is addition is a group. If I change this to multiplication what will happen? So, you can see. So, you can have a set with closure properties, but you cannot have a under multiplication product of 2 integers will give you an integer, but you will have problems with identity. Identity is also there, but inverse will not be there. So, this is one example which forms a group and further you can say that it is abelian because m 1 plus m 2 is same as m 2 plus m 1 to multiplication it is not a group because inverse is not there is that clear ok. So, it is abelian, but the set has infinite elements. So, it is an infinite number of elements. Other one is also set of complex numbers is a group under addition of complex numbers again you can show it to be abelian and infinite. This notation C minus 0 basically means it is a set of complex numbers excluding 0 that is the meaning of writing it as C minus 0 ok. C is the set of complex numbers complex numbers can also include 0, you remove the 0. So, the complex numbers can be plotted on a two dimensional plane is y axis and x axis all points on this two dimensional plane where every complex number z can be written as x plus i y you all know this right. I am going to remove this point out of this plane. So, this is removed this I am removing it. What is the advantage of removing it? Why am I removing it? 0 1 over 0 problems will not be there and then for this set without the 0 I can still show it to be an abelian group under multiplications ok. The earlier one was under addition. So, if you want to do multiplication you can use the space without having the origin showing pictorially the space and equivalently cancel it ok. Any other question on this? This is another example with complex entries. So, you can have infinite elements infinite number of elements and you can show that all the properties are satisfied. Is it satisfied? What is the group operation I am going to use? I am going to use matrix addition ok. The group operation is matrix addition. So, G 1 plus G 2 will give me this also belongs to that set of because A 1, B 1, C 1, D 1 are all complex entries and this will also belong to that set. What is an identity matrix here? Identity element, identity element is identity matrix inverse will be negative entries. So, it forms a group does it form an abelian or a non abelian? Addition, louder, abelian ok. So, it is abelian. Let us replace this group operation by matrix multiplication. All the properties are satisfied? What are the properties where you have problems? Inverse. If it is singular matrix then you are stuck. You have to make sure that they are non singular matrices whose determinant is not 0 then only you can find inverses ok. So, let us take it to be set of non singular matrices or determinant not 0 then it forms a group. Is it abelian or non abelian? So, matrix multiplication all of you know need not be commutative order can matter. Some cases A B can be B A, but not all elements in that. So, it is the first example with set of matrices whose determinant is not 0. So, that it is invertible can find inverses right. With matrix multiplication as the group operation it forms a non abelian group. So, I am hope by now you are all now familiar what I mean by a group. It is not just set you will have to give an operation and then under that operation all the four properties have to be satisfied. And then if on top of it if you have commutative property it is an abelian group four properties are satisfied it is a group. On top of it if it is commutative then it is abelian many times it will not satisfy commutative. The commutative should be satisfied all the elements not just some elements if all the elements are satisfying it is an abelian. So, we have seen the first three examples are commutative abelian group and the fourth one is also abelian group under addition. And then I asked what is the nature of the group if matrix multiplication replaces the matrix addition and we have already seen it will be a non abelian group provided they are non singular matrices or equivalently their determinant has to be non zero. So, this confining ourselves to the square incidentally I should also just take you back on this before this definition I had said discrete group is rotation of a square. That continuous group is where you can take a circle all the points on the circle under any rotation any small angle finite angle it will look exactly the same ok. So, this is where you will see that it will have many elements and this rotation angle is not a finite angle it is not discrete it can be continuous you can do r of theta can do r of theta you can do r of theta plus delta theta. So, the theta which is a rotation angle can be varying continuously also and this will again give you an r of some new angle which will involve rotation by 2 theta plus delta theta and so on ok. So, in that sense I would say that the theta is can have any value between 0 and 2 pi unlike your square where theta has to be only n times pi by 2 it is discrete whereas, this theta is continuous can be any in that sense the corresponding symmetry we call it to be a continuous. Is this set which involves rotation by the set is going to be infinite elements is it abelian or not abelian? Abelian right because I am moving the points on the circle there are infinite elements like the square which had only 4 elements, but it is still an abelian because I am going to do a rotation about the center of the circle and it is not going to give me anything fine ok. So, there is couple of things which I have still not defined for you what is a subgroup what is a multiplication table. So, let me just spend some time on that. So, we saw the rotation on a square ok. So, this one the symmetry group the group involves identity r pi by 2 r pi r 3 pi by 2 all with me. So, with some group operation which is just multiplying the rotation operation ok. What is a subgroup you have to find a subset you have to find a subset with again all the 4 properties can I find a subset here. So, this by this I mean subset of G which one will it be r pi. So, this one is what I call this this is what I call it as H let me write the H is a subset of G satisfying all the 4 properties. What is the inverse itself inverse identity is there associativity and closure property as trivial. So, it is still a satisfying the group property. So, we call this H as a subgroup. So, you can start doing this for other cases other examples you can see whether there are subgroups in it I am not really I gave you few examples for a group it starts seeing whether there are subgroups and then find out the subgroups which are sitting inside the bigger exactly like subsets inside sets ok and then we also can write the multiplication table ok. So, the multiplication table goes this way this is one thing which will show you all the properties. You write all the elements both along the row and along the column kind of a grid and then do the group operations between these elements. So, identity with identity is E identity with r pi by 2 is this r pi r 3 pi by 2. If you do r pi by 2 with identity is again r pi by 2 r pi by 2 with r pi by 2 will be r r pi and then this one will give you r 3 pi by 2 and r pi by 2 with r 3 pi by 2 is identity. What do you see in the second line as compared to the first line? What is the second line showing? The same four elements with some permutation is happening no repetition of any element each element in that set appears only once along a row or along a column of this multiplication table, but there is some permutation. So, that tells you that when you do this what is this? So, this is again a permutation. The last one are you all with me? So, this is what is called as a multiplication table. Typically it is useful if you have finite number of elements in a group ok. So, the number of elements in a group in any group G we call this as order of G and we denote it by. So, in this particular case this G has order there is the number of elements in that group. So, for that particular multiplication table order of the group is 4. So, it will be a 4 by 4 grid and each row and each column will have all the four elements in suitable permutation. So, that is what is the multiplication. So, I have gone through this examples. I have also said what is the order of a group by giving you the specific example where the number of elements is finite. If it is infinite the order will be very it will be infinite ok. We are going to confine ourselves to finite groups. Subgroups I have already said a subset satisfying all the four axioms of the group.