Projective space

A projective line (P ) is the set of radial lines in a projective plane. 

An n-dimensional projective space is obtained by starting with R and completing it by addition of its points at inhnity. 

Every pair of points in P lies on exactly one line. 

Every pair of lines in P intersects in exactly one point. 

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A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

Initially, the whole data set was analyzed by the linear PCA. By examining the behaviors of the process data in the projection spaces defined by small number of principal components, it... 

However, to determine the number of real pieces of information required to fix the projection from an M-dimensional space onto an /V-dimensional subspace spanned, not by the particular (occupied) basis in which P is diagonal, but by any basis of the subspace, it is necessary to subtract the numberof real parameters required to fix a particular basis in the /V-dimensional subspace from the total Kcy, such a number corresponds to the N2 real conditions that are necessary to fix a unitary transformation [11] in the subspace. But, as the phases ofthe eigenstates, < , are arbitrary as far as the physical state is concerned [4, 12], this latter number is reduced by N, the number of eigenstates belonging to the projection space. Hence, the number of independent real parameters in the unitary transformation which fixes the basis spanning the... 

Now we consider the case of projective space P. The Chow groups At(P ) and Ai(H3(Pd)) have already been determined in [Rossello-Xambo (2)]. 

In section 3.2 we consider the varieties of higher order data D X). Their definition is a generalisation of that of D X). We show that only the varieties of third order data of curves and hypersurfaces are well-behaved, i.e. they are locally trivial bundles over the corresponding varieties of second order data with fibre a projective space. In particular D X) is a natural desingularisation of. Then we compute the Chow ring of these varieties. As an enumerative application of the results of chapter 3 we determine formulas for the numbers of second and third order contacts of a smooth projective variety X C Pn with linear subspaces of P. ... 

Figure 3.49. Projection from a 2D periodic square lattice to obtain (on Xn) a 1D non-periodic crystal. The projection space Xn is inclined with an irrational slope (1 It) to the square lattice axes. The window W (the strip) parallel to Xn is chosen so that a structure with a physically reasonable density is obtained (Kelton 1995). The points contained in the window are projected.

The main classification methods for drug development are discriminant analysis (DA), possibly based on principal components (PLS-DA) and soft independent models for class analogy (SIMCA). SIMCA is based only on PCA analysis one PCA model is created for each class, and distances between objects and the projection space of PCA models are evaluated. PLS-DA is for example applied for the prediction of adverse effects by nonsteroidal anti-... 

C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces . Progress in Math. 3, Birakhauser, 1980. 

Mathematically, we collect the ambiguity of the phase factor into an equivalence relation (see Definition 1.3 of Section 1.7). In the current section we introduce the necessary equivalence relation and use it to define complex projective spaces. acquaint ourselves in some detail with the complex projective space P(C2). Finally, we show that linear transformations survive the equivalence. 

The set P( V) is sometimes called the projective space over V, complex projective space or, simply, projective space. 

More precisely, we can use stereographic projection to find an injective, surjective function from the projective space P(C2) to the sphere 5, via the plane of ratios. Stereographic projection is a function F from the xy-planc in R- into the unit sphere in R. We define... 

The projective space P(C2) has many names. In mathematical texts it is often called one-dimensional complex projective space, denoted CP (Students of complex differential geometry may recognize that the space PCC ) is onedimensional as a complex manifold loosely speaking, this means that around any point of (C ) there is a neighborhood that looks like an open subset of C, and these neighborhoods overlap in a reasonable way.) In physics the space appears as the state space of a spin-1/2 particle. In computer science, it is known as a qubit (pronounced cue-hit ), for reasons we will explain in Section 10.2. In this text we will use the name qubit because CP has mathematical connotations we wish to avoid. 

For each natural number n, there is a projective space P(C" ), also known as CP . Each element of PCC"" " ) is an equivalence class... 

However, the notion of a linear subspace descends to projective space. Definition 10.2 IfW is a linear subspace of V, we define... 

Not every linear-subspace-preserving function on projective space descends from a complex linear operator. However, when we consider the unitary structure in Section 10.3 we find an imperfect but still useful converse — see Proposition 10.9. 

At last, after several chapters of pretending that the state space of a quantum system is linear, we can finally be honest. The state space of each quantum system is a complex projective space. The reader may wish to review Section 1.2 at this point to see that while we were truthful there, we omitted to mention that unit vectors differing by a phase factor represent identical states. (In mathematics, as in life, truthful and honest are not synonyms.) In the next section, we apply our new insight to the spin state space of a spin-1/2 particle. 

The natural model for a spin-1/2 particle, a model that incorporates all the possible spin experiments, is the projective space We will wait... 

The reader familiar with the presentation of the state space of a spin-1/2 particle as S /T (i.e., the set of normalized pairs of complex numbers modulo a phase factor) may wonder why we even bother to introduce P(C2). One reason is that complex projective spaces are familiar to many mathematicians in the interest of interdisciplinary communication, it is useful to know that the state space of a spin-1/2 particle (and other spin particles, as we will see in Section 10.4) are complex projective spaces. Another reason is that in order to apply the powerful machinery of representation theory (including eigenvalues and superposition), there must be a linear space somewhere in the background by considering a projective space, we make the role of the linear space explicit. Finally, as we discuss in the next section, the effects of the complex scalar product on a linear space linger usefully in the projective space. 

If the kets label individual states, i.e.. points in projective space, and if addition makes no sense in projective space, what could this addition mean The answer lies with the unitary structure (i.e., the complex scalar product) on V and how it descends to P(y). If V models a quantum mechanical system, then there is a complex scalar product ( , ) on V. Naively speaking, the complex scalar product does not descend to an operation on P(V). For example, if v, w e V 0 and v, w Q v/e have u 2v but (v, w) 2 v, w) = 2v, w). So the bracket is not well defined on equivalence classes. Still, one important consequence of the bracket survives the equivalence orthogonality. 

So it does make sense to say that two equivalence classes, i.e., two points of projective space, are orthogonal. 

For example, consider a particle of spin 1/2. We can build a basis of the corresponding projective space by considering spin along the --axis. There are only two certain spin states, up and down. These are mutually exclusive if a particle is spin up, then it will not exit spin down from a --axis Stern-Gerlach machine, and vice versa. But is this set of states large enough Do either of these states have multiplicities In other words, is there some measurement that can distinguish between two pure spin-up particles, or between two pure spin-down particles The answer is no. As far as experiments have been done, any two z-spin-up (resp., spin-down) spin-1/2 particles are absolutely identical. So the list... 

Orthogonality in P(C2) is quite different from Euclidean orthogonality in three-space. In other words, although the projective space PCC ) can be thought of as the sphere as indicated in Figure 10.5, the two points [1 0] and [0 1], which are orthogonal as elements of the projective space, correspond to two points on the sphere that are antipodal, not orthogonal, in the Euclidean sense. 

Definition 10.5 Suppose V is a complex scalar product space with complex scalar product ( , >. Hfe define the absolute bracket on the projective space P(V)... 

In this section we have studied the shadow downstairs (in projective space) of the complex scalar product upstairs (in the linear space). We have found that although the scalar product itself does not descend, we can use it to define angles and orthogonality. Up to a phase factor, we can expand kets in orthogonal bases. We will use this projective unitary structure to define projective unitary representations and physical symmetries. 

The existence of spin-1/2 particles is evidence that the projective-space model is correct. For a description of the relevant experiments with Stern-Gerlach machines, see the Feynman Lectures [FLS, Vol. Ill, Chapter 6]. 

To extend this result to projective space of arbitrary finite dimension we will need the technical proposition below. Since addition does not descend to projective space, it makes no sense to talk of linear maps from one projective space to another. Yet something of linearity survives in projective space subspaces, as we saw in Proposition 10.1. The next step toward our classification is to show that physical symmetries preserve finite-dimensional linear subspaces and their dimensions. 

With Proposition 10.8 and the technical result Proposition 10.9 in hand, we are ready to classify the physical symmetries of complex projective spaces of arbitrary finite dimension. 

Consider a quantum system consisting of the spin state of two particles, one of spin 1/2 (a fermion) and one of spin 1 (a boson). How do we model the two of them together The (experimentally justified) prescription in quantum mechanics is to find a basis of states and build a projective space out of them, as we discussed briefly in Section 10.3. In this section we will show that the set... 

To model the two-particle system mathematically we need to find a mathematical projective space whose basis corresponds to the list of six states. We want more than just dimensions to match we want the physical representations on the individual particle phase spaces to combine naturally to give the physical representations on the combined phase space. The space that works is... 

To specify the final state of a measured particle, we need one more tool, orthogonal projection in projective space. We would like to consider the projectivization of n w, but since FT w is not necessarily invertible, we caimot apply Proposition 10.1. To evade this technical difficulty we restrict the domain of riiv. Recall the notation for set subtraction A B = x e A x B. ... 

Consider, for example, the measurement of the spin of a spin-1/2 particle via a Stem-Gerlach machine oriented along an arbitrary axis. Let [co ci] be the point in projective space corresponding to the positive axis of the Stem-Gerlach machine. Then a spin-up measurement corresponds to the onedimensional subspace VTup of spanned by (co, Ci) while a spin-down measurement corresponds (by Exercise 10.7) to the one-dimensional sub-... 

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