1 Dirac Notation for Vector Spaces

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1 Theoretical Physics Notes 2: Dirac Notation This installment of the notes covers Dirac notation, which proves to be very useful in many ways. For example, it gives a convenient way of expressing amplitudes in quantum mechanics, which are the complex numbers (which give the probability for something to occur upon being multiplied by their complex conjugate). It also lets us go smoothly from finite vector spaces (where matrix notation suffices) to the infinite-dimensional vector spaces of functions (where matrices would become infinite, sometimes needing a continuous index). So, in the following you will often see expressions in Dirac notation compared with the more familiar matrix notation used in linear algebra (and a few equivalents from tensor notation as well, when appropriate). 1 Dirac Notation for Vector Spaces We are used to vectors as being little arrows: objects with a magnitude and direction. Or, they may be represented as sets of several numbers, where the rules for addition and subtraction of vectors means that you must add (or subtract) the numbers kept at corresponding points in the set. It is more useful to think of these as merely special cases of more general vector spaces, where the rules for addition and subtraction, called the linear structure of the vector space, is fundamental to the definition. In other words, a vector space is anything where we can take two members of the space and form linear combinations of them: given vectors A and B and two numbers (real or complex) α and β, if it makes sense that α A + β B is also a vector (which we may then call αa+βb ), then that is the very definition of a vector space (real or complex, depending upon the numbers used). This clearly includes our component lists and little arrows, but also other examples of vast importance to theoretical physics, in particular vector spaces of functions obeying linear differential equations. This means vector notation can be used in Maxwell s equations, and also of course for abstract Hilbert space which is used to represent quantum systems (for which the Dirac notation already being used above was invented). Given any vector space, there always exists a dual vector space, whose objects are linear maps from the vectors to ordinary numbers (real or complex). Dirac notation makes clear the distinction between these objects, called covectors or one-forms, and the regular vectors. If you think of vectors as little arrows, then covectors are linear functions on the space where each arrow starts at the origin (as explained below). Dirac notation is often called braket notation, because in it, a number is represented by a bracket, given by the number produced by a covector acting on a vector. Inner products are often described as being generalizations of the ordinary dot product familiar from vector analysis, but that s actually a bit misleading. Inner products are actually a generalization of something more basic than dot products: a dot product 1

2 is actually an inner product together with a map which allows us to identify specific vectors with specific covectors (and vice-versa), as we ll see later below. 2 Summary of Dirac Notation After each object, the equivalents from linear algebra (LA) and in tensor notation (T) are given in parentheses, with some definitions to follow after the summary.... denotes a vector (LA: column vector; T: type (1,0) tensor whose components have one upstairs index, e.g. a i )... denotes a covector, also known as a one-form (LA: row vector; T: type (0,1) tensor with one down index on components) denotes a number formed by the inner product of a covector with a vector (LA: row vector on left, matrix product with column vector on right; T: summation over index of product of covector components with same-index vector components) denotes an outer product of a vector with a covector (LA: column vector on left, matrix product with row vector on right, giving a matrix; T: set of all products of a single covector component with a single vector component with no sum over indices) 3 Some Definitions and Examples 3.1 Covectors are Different than Vectors, Usually Given any vector space, the space of covectors (one-forms) is defined to be the space of linear maps from the vectors to regular numbers (where the numbers can be real or complex). That is, a covector is a machine which is ready to eat a vector and spit out a number, and to do so in a linear way. This means that if a covector φ is fed a linear combination of vectors, the number which it produces obeys φ αa+βb = α φ A +β φ B., for all vectors A and B and all numbers α and β. This means that the zero covector is a perfectly acceptable (but dull) covector, and in fact the space of covectors is somewhat confusingly a vector space in its own right! So covectors are always defined with some respect to some specific vector space which we have in mind. (Fortunately, when a space of covectors is considered to be a vector space of its own, their covectors are members of the original vector space, so we never have too many vector spaces to worry about at one time.) In two dimensions, if we think of a vector as an arrow with its magnitude and direction, then a covector is given by something which looks a bit like a ruler: a set of tick-marks, evenly spaced. Looking closer, the tick-marks are actually little parallel lines, each perpendicular to the edge of the ruler. For the covector/one-form, we should think of these lines as extending infinitely far in each direction. But just as changing a vector s direction gives a different vector, it s only fair to consider a covector as having its specific orientation, and if we 2

3 change that orientation then it s a different covector. In other words, a ruler s tick marks and the way it is pointing are both needed before we know which covector it represents. Inner products are then easy to picture: to get the inner product of a vector with a covector, count the total number of lines of the covector that the arrow representing the vector intersects. If the vector is mostly parallel to the ruler, count the number as positive. If the vector is mostly oppositely oriented to the ruler, count the number as negative. And if the vector is perpendicular to the ruler by being parallel to the tick-mark lines, count the number as zero. As an exercise, you should verify that the property of acting linearly on vectors, which is built into the definition of a covector, requires that the ruler s ticks must be extended into parallel lines, as described above. In three dimensions, there are two independent directions perpendicular to the line formed by the very edge of the ruler, so a covector is made up of not evenly spaced parallel lines, but rather evenly spaced parallel planes, going out infinitely far in every direction perpendicular to the ruler s edge. Generally, in n dimensions a covector is made of evenly spaced n 1 dimensional spaces (R n 1 ), called hypersurfaces because they generalize a surface in the 3-D case. So if a vector is like an arrow, a covector is like a ruler, capable of giving out a number when fed an arrow. To use a ruler properly, you should line up its edge parallel to the arrow v v you want to measure. More importantly, you keep the tick marks perpendicular to v. The ruler is now the covector representation of v, or v! It is a unit covector; recall from the previous notes that units must be specified in order to turn dimensionful quantities into numbers, and the spacing of the tick marks on the ruler defines one unit. There are many such unit covectors, but lining it up with the direction of v produces v. The inner product v v = v, exactly as should be produced when a ruler is used to measure a length. So much for the arrow interpretation. If a vector is thought of as a column vector in linear algebra, then a covector is just a row vector: matrix multiplication is linear and so the inner product will automatically satisfy the necessary linear property. Again, these are clearly different objects than the column vectors. So, which is the vector and which is the covector, in the ordinary dot product of two vectors? And why wasn t this emphasized way back when you first took a vector analysis course? It turns out that in some situations, we can identify vectors with covectors. These include the enormously important cases of ordinary Euclidean space in an orthonormal vector basis, and also Hilbert space in quantum mechanics in those, you may be sloppy about which is which. But they must always transform oppositely under changes of coordinate system (which is the tensor analysis viewpoint). So it s best to think of one vector in a dot product (w v) either one, say v as the vector, and the other vector and the dot are then the covector (here, w, which explains the strange notation introduced above). In linear algebra, non-orthonormal vector bases show up as a metric, which is a matrix placed in between the components of w, written as a row vector, 3

4 and those of v, written as a column vector. The covector is in that case not the row vector made of the components of w, but rather is that row vector multiplied by the matrix representing the metric on its right (which makes a new row vector). So the metric matrix represents the dot in the notation, and its presence ensures that covectors transform the right way under changes of coordinates to leave inner products alone. Again, it should be emphasized that a covector and vector inner product never needs any metric between them: a dot is only required to take an inner product of a vector with another vector. 3.2 Inner Products There are many different contexts in mathematical physics where inner products between a vector space and its dual space occur, and so Dirac notation can be useful. Mathematicians call this type of inner product bilinear, meaning that it is linear on the vector space and on the dual space (which, you will recall, is also a vector space itself). In Dirac notation, bilinearity looks like w 1 +αw 2 v 1 +βv 2 = w 1 v 1 +α w 2 v 1 +β w 1 v 2 +α β w 2 v 2. Notice that we take the complex conjugate of complex numbers which appear as labels in the dual space, as α in the example above. When the number appears outside the covector label, though, we write the conjugate explicitly: w 1 +αw 2 = w 1 +α w 2 and so you ll note that for self-consistency we d better have w 1 +α w 2 = w 1 +α w 2. The complex conjugate rule, although it may be confusing at first, is very convenient: when it makes sense to use the same labels on covectors as on vectors (as with orthonormal bases in finite dimensions, or in Hilbert space for quantum mechanics), we have a positive real inner product when a vector is dotted with itself : v 1 + αv 2 v 1 + αv 2 0. This is, of course, the extension of the rule alluded to above in linear algebra on complex vector spaces: we take the conjugate of the row vector before multiplying by the column vector. (The quotes are to remind you that really, we are forming the covector which goes canonically with the vector being labeled: the row vector is really a covector, ready to give us a number when fed a column vector.) Thisformsourfirstexampleofaninnerproduct: thefamiliaronefromlinear algebra in a finite-dimensional vector space. For it, we define the covector by the complex conjugate transpose, also known as the Hermitian conjugate, of the column vector which shares its label, and then multiply by the other vector. In short, w v = wiv i = w T v = w v i Going across in this from left to right, we have first the Dirac notation, and then tensor notation (in which v i stands for the number in row i of the column vector 4

5 representing v), and finally matrix notation in both the third and last versions. In the matrix versions, matrix multiplication of the objects is understood (and written out explicitly in the tensor notation before). For matrices, T stands for transpose: you switch the row and column indices of any matrix to take its transpose, so the transpose of a column vector produces a row vector. The final version introduces dagger notation: the dagger ( ) is short for complex conjugate transpose in finite dimensions (and adjoint in infinite dimensions, defined when we get to linear operators). Again, you ll notice that covectors and vectors are genuinely different, even when it makes sense to use the same labels for them. In fact, it is true in general that w v = v w when we exchange the vectors and covectors (whenever it makes sense to do so in their labels), and so we get a different number unless the number happens to be real. Our next examples are extremely important ones: the vector spaces of functions on some fixed interval, area, or volume. We define the dual space of covectors by taking the complex conjugate of the function, and integrating over the domain: g f = g (x)f(x)d n x where the integration is over the whole domain in question, whose dimension is written as n above. Notice that we can use this to define a norm, or length, of these vectors, whenever the integral of f 2 over the domain isn t infinite, because it gives a positive real number whenever f is nonzero over some portion of the domain: we can define the length by f = f f. (Mathematicians call this the L 2 norm. ) In fact, this is often used to define the vector space more precisely. As we will see in the discussion of linear operators acting on functions, it is crucially important to specify the boundary conditions of the functions at the edges of the domain. One such specification very common in quantum mechanics is that the function is normalizable : its norm cannot be infinite. This does indeed form a vector space, as you can check: f +g will be finite whenever f and g both are. This definition even works when the domain is infinitely large (such as the whole real line). The all-important boundary condition comes from physical considerations: in quantum mechanics, setting the overall probability of finding a particle somewhere on the real line equal to 1 makes use consider wavefunctions of norm 1 only. (Careful, as the functions of norm 1 do not form a vector space! Add one such to itself and you will see why.) When the domain is finite, boundary conditions become even more important, and again their specification usually comes from physical considerations. For instance, the shape of a violin string which is clamped at both ends leads one to consider functions on a line segment which go to zero at both endpoints, called homogeneous Dirichlet boundary conditions : for example, y(a) = 0 and y(b) = 0 would be allowed functions for y(x) between x = a and x = b. Adding two such functions produces another, so we have a vector space. On the other hand, replacing the condition y(b) = 0 with y(b) = 1 (which is an inhomogeneous Dirichlet boundary condition) do not give a vector space: add two such together, and you ll have y total (b) = 2, so it isn t again an allowed function. This 5

6 doesn t mean that vector spaces are useless in this case; if we agree to subtract away some fixed reference function (such as the straight line between 0 at a and 1 at b) from every function we consider, the resulting functions will be zero at both ends, and will again form a vector space. Before moving on to another bracket product example, a couple of remarks about this integration product, and vector spaces of functions in general, are in order. First, vector spaces of functions on an interval are always infinitedimensional: given any finite list of functions, we can always find another one which isn t a combination of those already in the list. (Note that we refer here not to the dimension n of the domain of the functions, which is finite, but rather the abstract space of functions themselves is what is infinite: there are a lot of possible functions, all linearly independent of each other.) Second, the dual space can actually be larger than the original vector space in infinite dimensions (in finite dimensions, dual spaces always have the same dimension as the vector space they are dual to). For example, if our vector space consists of continuous functions on a line, the dual space will not only include such functions but will also include distributions : Dirac delta functions, which produce the value of f at a single point when their inner product integral is taken with some f(x). On the other hand, the dual of the dual space (our original vector space) does not have Dirac deltas in it, because the integral of a Dirac delta squared does not return a number (in fact, it does not make sense mathematically). Finally, it is very helpful to consider this example as being essentially the same asthefirstexample, wheretheinnerproductwasasumoverafinitenumber of vector and covector components. (We ll consider just n = 1 functions on a line for the discussion here.) The index i has simply become continuous, and turned into the label x. Instead of having one number for each index i (an n-dim. vector), we have one for each value of x (a function). In the inner product, the sum over i becomes an integral over dx, and then the analogy is complete. A third example of an inner product which shows up in theoretical physics, but looks quite different at first than those above, is that defined on certain vector spaces of matrices, where the matrices themselves are the vectors. A convenient inner product is to define A B = Tr(A B), where Tr denotes the matrix trace (sum of the diagonal elements) of the matrix product of the Hermitian conjugate of A with B. (Note that this does not even require that A and B be square matrices; just that they both have the same number of rows and columns. If they are not square, then Tr(BA ) defines a different inner product, but is identical when they are square.) For diagonalizable square matrices, this definition is very convenient because it does not change when we make a unitary change of basis: replacing A by U 1 AU and B by U 1 BU leaves their inner product unchanged if U is unitary (U 1 = U ), as you should check. We can also use this to define the norm of a matrix: for diagonalizable matrices, A A will return the sum of the complex-squared eigenvalues of A, which is never negative. Often, the inner product will be defined with an extra 1/n accompanying the trace for n by n matrices; this gives the n-dimensional unit matrix a norm of 1. 6

7 3.3 Outer Products are Linear Operators As defined above, an inner product is a bracket, with the bra covector and the ket vector sandwiched together to give a number. What, then, is an outer product, where the ket (... ) is to the left of the bra (... )? In linear algebra, this would correspond to the matrix product of a column vector on the left with a row vector to the right. Instead of producing a number (as a 1 by 1 matrix), this instead produces a matrix (n by n, where n is the dimension of the vector space). A matrix is a linear operator: it acts on column vectors to give other column vectors, and does so linearly. It can also act on row (co)vectors (multiplying the matrix on the right) to give other row (co)vectors, and that operation is also linear (doubling the input covector doubles the output covector). As the Dirac notation immediately suggests, this interpretation extends to infinite-dimensional vector spaces: an outer product is a linear operator, which can act linearly on the vectors and also linearly on covectors. The anti-sandwiched bra and ket in an outer product can operate on a ket to the right, where the inner product it forms with the bra part of the outer product produces a number. That number then multiplies the ket part of the outer product, and that ket times the number formed becomes the result of the operation. Similarly, the ket part of the outer product is ready to form an inner product with a bra covector on the left, producing a number to multiply the bra from the outer product. Of course, either result can then operate on yet another vector (from the dual space of the result of the first operation) to produce an ordinary number. In other words, an outer product can operate on both a bra (on the left) and a ket (to the right) to produce just a number (the product of the two inner products created by this operation). One simple but important example of an outer product is that of a projection operator onto a single one-dimensional subspace of the original vector (or covector) space. To create this, suppose that we have chosen a basis for our vector space (say, { φ i }, where the index i runs over all members of the basis this means that each member is linearly independent of the others, and so cannot be written as a linear combination of them, and that any vector can be written uniquely as a linear combination of basis vectors). We can always then create the canonically dual basis of covectors (which we ll use the same labels for). These are constructed to satisfy φ i φ j = δ ij : each basis covector is orthogonal to all basis vectors except the one which shares its label, and it has an inner product of 1 with that one. Then the projection onto the one-dimensional subspace spanned by the first basis vector, for instance, is given by P 1 = φ 1 φ 1. When it acts on a vector, it leaves unchanged that part of the vector which is a multiple of φ 1, and removes the rest of the vector (and likewise for covectors). For an ordinary example, suppose that we are in regular 3-dimensional space, and we re using the usual Cartesian basis x,ŷ, and ẑ. These will still come in two types each: once as a basis vector and once as a basis covector. Then the projections onto the second basis vector is given by P 2 = ŷ ŷ. When acting 7

8 upon an arbitrary vector v, written as v = v x x + v y ŷ + v z ẑ, it returns just P 2 v = v y ŷ. This represents the shadow of v in just the y direction. Projection operators have many nice features, some of which we ll summarize in an upcoming handout on linear algebra. Repeated application of a projection operator doesn t do anything after the first projection, so they satisfy P 2 = P (asiseasytocheckwiththeaboveexamples). Thisisalsotrueofprojectionsinto subspaces of dimension higher than 1, which are created by adding up several different one-dimensional projection operators (and these also are themselves when squared, as you can easily check). One of the most important such is a summation over the entire set of projection operators: this doesn t change anything, so it is an identity operator: i P i = I, which is called a completeness relation. Just like general projections can be written as a sum of outer products, so can any general linear operator also can be written as a linear combination of outer products. So, outer products are very useful in general, as providing a basis for linear operators. An example (using familiar 3-dimensional space once again) is given by a reshuffling of the basis vectors into others: consider the operator R = ŷ x + ẑ ŷ + x ẑ. It is easy to see that when this acts on a vector, it simply relabels the x, y, and z components cyclically. So, it is a rotation about the axis parallel to the vector (x,y,z) = (1,1,1) by 120 degrees. 8