Complex projective space

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. 

Suppose V is a complex scalar product space used in the study of a particular quantum mechanical system. (For example, consider V = L (R j, the space used in the study of a mobile particle in R. ) If v and w are nonzero vectors in V, and if there is a nonzero complex number A, such that v = kw, then V and w correspond to the same state of the quantum system since v = Xw, we have 

If we want to have a mathematical space in which each point corresponds to exactly one state of the quantum mechamcal system, we must construct a space of equivalence classes. 

Definition 10.1 Suppose V is a complex vector space. We define the projec-fivizafion of V by 

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

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

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. 

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 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. 

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. 

D. G. Kendall, Bull. London Math. Soc. 16, 81 (1984). Shape Manifolds, Procrustean Metrics, and Complex Projective Space. 

Kendall, D.G. (1984) Shape-manifolds, procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society, 16 81-121. 

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. 

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. 

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. 

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. 

We can discretize the model, in this case the anomalous conductivity distribution, Ad (r), by introducing a set of basis functions, V l (r), V 2 (r), / /v (r) in the finite dimensional Hilbert space M, which is a subspace of the complex Hilbert space Mo M/v C Mo- Let us approximate the anomalous conductivity by its projection over the basis functions ... 

The isotropic lines of a projective identity mapping define the local complex Minkowski space of special relativity directly. By taking the circular points at infinity into account the global projective space of general projective relativity is obtained. No other topology reveals the transition from special to general relativity as such a simple consequence of curved space-time. 

The real projective space RP can be represented as a CW complex with one cell in each dimension from 0 to n. This cell structure is the Z2-quotient, with respect to the antipodal map, of the cell structure on the sphere S , which we described in (l)(b) above. 

Systems Engineering is a worldwide accepted tool for the design of complex projects. It is used in many sectors of industry such as aero and space technology, telephone systems, defence industry and computer technology. Also for infrastructure works this method has proven its credibility. 

SONNIA can be employed for the classification and clustering of objects, the projection of data from high-dimensional spaces into two-dimensional planes, the perception of similarities, the modeling and prediction of complex relationships, and the subsequent visualization of the underlying data such as chemical structures or reactions which greatly facilitates the investigation of chemical data. 

The procedure of DG calculations can be subdivided in three separated steps [119-121]. At first, holonomic matrices (see below for explanahon) with pairwise distance upper and lower limits are generated from the topology of the molecule of interest. These limits can be further restrained by NOE-derived distance information which are obtained from NMR experiments. In a second step, random distances within the upper and lower limit are selected and are stored in a metric matrix. This operation is called metrization. Finally, all distances are converted into a complex geometry by mathematical operations. Hereby, the matrix-based distance space is projected into a Gartesian coordinate space (embedding). 

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