Vector Spaces in Physics 8/11/2010 2 - 1 Chapter 2. The Special Symbols ijδand ijkε, the Einstein Summation Convention, and some Group Theory Working with vector components and other numbered objects can be made easier through the use of some special symbols and techniques. We will discuss two symbols with indices, the Kronecker delta symbol and the Levi-Civita totally antisymmetric tensor. We will also introduce the use of the Einstein summation convention. References. Scalars, vectors, the Kronecker delta and the Levi-Civita symbol and the Einstein summation convention are discussed by Lea [2004], pp. 5-17. A. The Kronecker delta symbol, ijδ. This symbol has two indices, and is defined as follows: symbol deltaKronecker 3,2,1,,,1,0==≠= jijijiijδ (2-1) Here the indices i and j take on the values 1, 2, and 3, appropriate to a space of three-component vectors. The same definition could in fact be used in a space of any dimensionality. We will now introduce new notation for vector components, numbering them rather than naming them. [This emphasizes the equivalence of the three dimensions.] We will write vector components as {}x y z(A ,A ,A ) , 1,3iA i→ = (2-2) We also write the unit vectors along the three axes as {}{ }ˆˆ ˆˆ, , , 1,3ii j k e i→ = (2-3) The definition of vector components in terms of the unit direction vectors is 3,1,ˆ=•≡ ieAAii (2-4) The condition that the unit vectors be orthonormal is ijjieeδ=•ˆˆ (2-5) This one equation is equivalent to nine separate equations: 1ˆˆ=• ii, 1ˆˆ=• jj , 1ˆˆ=• kk, 0ˆˆ=• ji , 0ˆˆ=• ij , 0ˆˆ=• ki , 0ˆˆ=• ik, 0ˆˆ=• kj , 0ˆˆ=• jk !!! [We have now stopped writing "i,j=1,3;" it will be understood from now on that, in a 3-dimensional space, the "free indices" (like i and j above) can take on any value from 1 to 3.] Example: Find the value of kjˆˆ• obtained by using equation (2-5). Solution: We substitute 2ˆe for jˆ and 3ˆe for kˆ, giving 0ˆˆˆˆ2332==•=•δeekj , correct since jˆ and kˆ are orthogonal.Vector Spaces in Physics 8/11/2010 2 - 2 B. The Einstein summation convention. The dot product of two vectors A and B now takes on the form ∑==•31iiiBABA. (2-6) This is the same dot product as previously defined in equation (2-13), except that AxBx has been replaced by A1B1 and so on for the other components. Now, when you do a lot of calculations with vector components, you find that the sum of an index from 1 to 3 occurs over and over again. In fact, occasions where the sum would not be carried out over all three of the directions are hard to imagine. Furthermore, when a sum is carried out, there are almost always two indices which have the same value - the index i in equation (3-6) above, for example. So, the following practice makes the equations much simpler: This sounds a bit risky, doesn't it? Will you always know when to sum and when not to? It does simplify things, though. The reference to tensor indices means indices on elements of matrices. We will see this convention is expecially adapted to matrix multiplication. So, the definition of the dot product is now iiBABA =•, (2-7) the same as equation (3-6) except that the summation sign is omitted. The sum is still carried out because the index i appears twice, and we have adopted the Einstein summation convention. To see how this looks in practice, let's look at the calculation of the x-component of a vector, in our new notation. We will write the vector A, referring to the diagram of figure 1-11, as iizyxAeAkAjAiAˆˆˆˆ=++=, (2-8) where in the second line the summation over i = {1,2,3} is implicit. Now use the definition of the x-component, 11ˆeAAAx•≡=. (2-9) The Einstein Summation Convention. In expressions involving vector or tensor indices, whenever two indices are the same (the same symbol), it will be assumed that a sum over that index from 1 to 3 is to be carried out. This index is referred to as a summed index. An index which only occurs once in a term of an expression is referred to as a free index, and is not summed.Vector Spaces in Physics 8/11/2010 2 - 3 Combining (2-8) and (2-9), we have 1 11111ˆˆ ˆ( )ˆ ˆ( )i ii ii iA A ee A eA e eAAδ= •= •= •== (2-10) In the next to last step we used (2-5), the orthogonality condition for the unit direction vectors. Next we carry out one of the most important operations using the Kronecker delta symbol, summing over one of its indices. This is also very confusing to someone seeing it for the first time. In the last line of equation (2-10) there is an implied summation over the index i. We will write out that summation term by term, just this once: 3132121111δδδδAAAAii++= (2-11) Now refer to (2-1), the definition of the Kronecker delta symbol. What are the values of the three delta symbols on the right-hand side of the equation above? Answer: δ11 = 1, δ21 = 0, δ31 = 0. Substituting these values in gives 13213132121111001AAAAAAAAii=∗+∗+∗=++=δδδδ (2-12) Notice that the summation over the shared index i has the effect of transfering the index 1 from the delta symbol to A. C. The Levi-Civita totally antisymmetric tensor. The Levi-Civita symbol is an object with three vector indices, (), 1, 23; 1, 2,3; 1, 2,3 Levi-Civita Symbolijki j kε= = = (2-13) All of its components (all 27 of them) are either equal to 0, -1, or +1. Determining which is which involves the idea of permutations. The subscripts (i,j,k) represent three numbers, each of which can be equal to 1, 2, or 3. A permutation of these numbers scrambles them up, and it is a good idea to approach this process systematically. So, we are going to discuss the permutation group. Groups. A group is a mathematical concept, a special kind of set. It is defined as follows: A group is a set of objects {}, , ,...A B Cwith multiplication of one member by another defined, closed under multiplication, and with the additional properties: (i) The group contains an element I called the identity, such that, for every element A of the group, AI IA A= = (2-14) (ii) For every element A of the group there is another element B, such thatVector Spaces in Physics 8/11/2010 2 - 4 AB BA I= =. (2-15) B is said to be the inverse of A: 1A B−=. (2-16) (iii) Multiplication must be associative: ()()A BC