Natural complex scalar product

For example, for any natural number n there is a natural complex scalar product on the n-dimensional complex vector space C" defined by... 

This is the group of determinant-one linear operators on the complex vector space preserving the natural complex scalar product ( , ) dehned in Section 3.2. Note that... 

If both factors are unitary representations, then so is the tensor product. If both V and V have complex scalar products defined on them, then there is a natural complex scalar product on the tensor product V 0 V of vector spaces. Specifically, we define... 

Proposition 5.11 Suppose (G, V, p) is a finite-dimensional unitary representation with character /, Then the character of the dual representation G, V, p ) is X - (Recall that x denotes the complex conjugate of the C-valuedfunction xf Fitt thermore, (G, V, p is a unitary representation with respect to the natural complex scalar product on V. ... 

Exercise 5.21 Suppose V is a complex vector space. Show that V = Hom( V, C). Now suppose that V has a complex scalar product. Do the natural complex scalar products induced on V (defined in Exercise 3.19) and Hoin( V, C) (defined in Exercise 3.20) agree ... 

Exercise 5.22 (Used in Proposition 11.1) Suppose V and W are complex scalar product spaces. Recall (from Exercise 3.19, Exercise 3.20 and Equation 5.2) the natural complex scalar products ( , >Hom(v iv)... 

Recall from Section 5.3 that the natural complex scalar product on a tensor product is obtained by multiplying the individual complex scalar products of the factors. Letting ( , -jj and ( , ->2 denote the complex scalar products on the factors Vk and V, respectively, we find... 

The natural mathematical setting for any quantum mechanical problem is a complex scalar product space, dehned in Dehnition 3.2. The primary complex scalar product space used in the study of the motion of a particle in three-space is called (R ), pronounced ell-two-of-are-three. Our analysis of the hydrogen atom (and hence the periodic table) will require a few other complex scalar product spaces as well. Also, the representation theory we will introduce and use depends on the abstract nohon of a complex scalar product space. In this chapter we introduce the complex vector space dehne complex scalar products, discuss and exploit analogies between complex scalar products and the familiar Euclidean dot product and do some of the analysis necessary to apply these analogies to inhnite-dimensional complex scalar product spaces. 

Complex scalar products arise naturally in quantum mechanics because there is an experimental interpretation for the complex scalar product of two wave functions (as we saw in Section 1.2). Students of physics should note that the traditional brac-ket notation is consistent with our complex scalar product notation—just put a bar in place of the comma. The physical importance of the bracket will allow us to apply our intuition about Euclidean geometry (such as orthogonality) to states of quantum systems. 

This is a special case of a more general construction. If there is a complex scalar product on V. then there is a natural linear transformation r V V defined by... 

Consider a complex scalar product space V that models the states of a quantum system. Suppose G is the symmetry group and (G, V, p) is the natural representation. By the argument in Section 5.1, the only physically natural subspaces are invariant subspaces. Suppose there are invariant subspaces Gi, U2, W c V such that W = U U2. Now consider a state w of the quantum system such that w e W, but w Uy and w U2. Then there is a nonzero mi e Gi and a nonzero M2 e U2 such that w = ui + U2. This means that the state w is a superposition of states ui and U2. It follows that w is not an elementary state of the system — by the principle of superposition, anything we want to know about w we can deduce by studying mi and M2. 

Proposition 9.3 For any nonnegative integer n, the natural map (restriction) from IHI" to is an isomorphism of complex vector spaces. The set 3 4 spans the complex scalar product space 7, (5 ). 

It is natural to ask which operations descend from V toP(V). IsP(V) a complex vector space Usually not. If V has a complex scalar product, does P( V) have a complex scalar product No. But. as we will see in this section, a complex scalar product on V does endow P(V) with a useful notion of orthogonality. Furthermore, using the complex scalar product on V we can measure angles in P( V). At the end of the section we apply this new technology to the qubit P(C"). 

Proposition 10.10 Suppose n is a natural number and ( , ] is the standard complex scalar product on C . Suppose S P(C") —> P(C") is a physical symmetry. Then there is a unitary operator T C" C" a function k, equal to either the identity or the conjugation fi me lion, such that... 

T natural isomorphism from a complex scalar product space to its... 

Exercise 2.4 Show that for any natural number n, the Cartesian product C is a complex vector space of dimension n. Then show that C with the usual addition but with scalar multiplication by real numbers only is a real vector space of dimension 2n. 

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