 It is impossible to imagine theoretical physics without the Kroneka delta. You will encounter this relatively simple yet powerful tensor practically in all fields of theoretical physics. For example, it is used to write long expressions more compactly and to simplify complicated expressions. In combination with the Levi-Civitta tensor, the two tensors are very powerful. That's why it's worth understanding how the Kroneka delta works. Kroneka delta ij is a small Greek letter delta which yields either 1 or 0 depending on which values its two indices i and j take on. The maximal value of an index corresponds to the considered dimension, so in three-dimensional space i and j run from 1 to 3. Kroneka delta is equal to 1 if i and j are equal and Kroneka delta is 0 if i and j are not equal. Let's make some examples. Delta 11 is equal to 1 because the indices are equal. Delta 23 is 0 because the indices are not equal. A times delta 33 is equal to A because the indices are equal, so the Kroneka delta is 1 and delta 23 times delta 22 is equal to 0 because the first delta yields 0 and the second delta yields 1, but 0 times 1 is still 0. In order to represent an expression like this or an expression like this compactly, we agree on the following summation convention rule. We omit the sum sign, but keep in mind that if two equal indices appear in an expression, then we sum over that index. Another advantage of the sum convention in addition to compactness is formal commutativity. For example, you may write down the expression like this in a different order as you wish for example like this. This might help you see what can be shortened or simplified further, but be careful. There are exceptions for example with a differential operator delta j which acts on a successor. You can't move something before the derivative that is supposed to be differentiated, so you should be careful with operators in index notation. Now let's learn some important rules using Kroneka delta. Rule number one, indices i, j may be interchanged. Delta i, j is equal to delta j, i. Why is that? According to the definition, if the indices i and j are equal, then delta i, j is equal to 1, but then also delta j, i is equal to 1. And if the indices are unequal, you have delta i, j is equal to 0, and delta j, i is equal to 0 as well. So you can see Kroneka delta is symmetric. Rule number two, if the product of two or more Kroneka deltas contains a summation index, like in this case j, then the product can be shortened such that the summation index j disappears. Delta i, j times delta j, k is equal to delta i, k. So instead of writing two deltas, you can just write delta i, k. We say the summation index j is contracted. Let's do an example to understand it better. Consider delta k, m times delta m, n. The summation index here is m, so you can eliminate it by contracting it and you get delta k, n. Next example, delta i, j times delta k, j times delta i, n. Here you have two summation indices i and j. So in principle, you can eliminate both of them. From the first rule, you know that Kroneka delta is symmetric. So you can swap k and j in delta k, j and then contract the index j. You get delta i, k times delta i, n. And then you contract the summation index i. The simplified result is delta k, n. Remember that the contraction order is not important here. You could have contracted i first instead of contracting j first. In both cases, you get the same result, delta k, n. So which wave simplification you take does not matter. Rule number three. If the index in a, j also occurs in delta j, k, then the Kroneka delta disappears and the factor a, j gets the other index k. a, j times delta j, k is equal to a, k. This rule is basically another case of index contraction. This rule tells you that you can also contract summation indices that don't have to be carried by Kroneka delta. Let's make another example. Consider gamma, j, m, k times delta, n, k. The summation index is k, so you can eliminate it. The result is gamma, j, m, n. Rule number four. If j runs from 1 to n, then delta, j, j is equal to n. Why is that? According to the summation convention, the summation is carried out over j here. So delta, j, j is equal to delta 11 plus delta 22 and so on, up to n. And each Kroneka delta yields 1 because the index values are equal. So 1 plus 1 plus 1 and so on results in n. If, for example, like in our case, j runs from 1 to 3, then delta, j, j will be equal to 3. Consider a three-dimensional vector v with the components x, y, and z. You can represent this vector v in an orthonormal basis as follows. Here, ex, ey, and ez are three basis vectors which are orthogonal to each other and normalized. In this case, they span an orthogonal three-dimensional coordinate system. The Kroneka delta needs vectors written in index notation. Here we do not denote the vector components with different letters x, y, z, but we choose one letter, here the letter v, and then number the vector components and the basis vectors consecutively. The vector components are then called v1, v2, and v3. One of the advantages of index notation is that this way you will never run out of letters for the vector components. Just imagine a 50-dimensional vector. They are not even that many letters to give each component a unique letter. Another advantage of the index notation is that by numbering the vector components in this way, you can use the sum sign to represent the basis expansion more compactly. It becomes even more compact if we omit the big sum sign according to the summation convention. Look how compact the vector v can be represented in the basis. v is equal to vj times ej. Here is you know with sum over index j. Whether you call the index j, i, or k, or any other letter, is of course up to you. Now that you know how a vector is represented in index notation, we can analogously write the scalar product of two vectors a and b in index notation. For this we use the just learned index notation of a vector. aiei, scalar product bj, ej. In index notation you may sort the factors as you like. This is the advantage of index notation where the commutative law applies. Let's take advantage of that and put parentheses around the basis vectors to emphasize their importance in introducing the chronica delta. The basis vectors ei and ej are orthonormal. Recall what the property of being orthonormal means for two vectors. Their scalar product yields either 1 if i and j are equal, or it yields 0 if i and j are not equal. Doesn't this property look familiar to you? The scalar product of two orthonormal vectors behaves exactly like chronica delta. Therefore replace the scalar product of two basis vectors with a chronica delta. So a scalar product b is equal to ai times bj times delta ij. If you remember the third rule, you can contract the summation index j if you want. So the scalar product becomes ai times bi. And you get exactly the definition of the scalar product, where the vector components are summed component-wise. We can write out the double summation over i and j just for practice. In other words, we have to go through all possible combinations of the indices i and j. So we get a1 b1 delta11 plus a1 b2 delta12 and so on. Because of the definition of chronica delta, only three components of 9 in total are not 0, where i equals j. So you may omit all summands with unequal indices. Using the definition of chronica delta, you get the scalar product you are familiar with.