Unitary state

The boundaries drawn between British and Dutch spheres in what the British called the Malay World were arbitrary. The peoples and histories in the space that came to be Malaysia were not fundamentally different from those that came to be Indonesia save for their more recent, immigrant quality. Indonesia had by far the more intractable assemblage of ethnies rooted in distinctive histories, languages and literatures. Yet it is Malaysia that has the complex federal system, while Indonesia remains, with China, the world s largest experiment in organising exceptional cultural diversity through a unitary state. 

The development of an Observer can allow a person considerable access to observing different identity states. An outside observer can often clearly infer different identity states, but a person who has not developed the Observer function well may never notice his many transitions from one identity state to another. Thus ordinary consciousness, or what society values as "normal" consciousness, may actually consist of a large number of d-SoCs, identity states. But the overall similarities between these identity states and the difficulty of observing them, for the reasons discussed above, lead us to think of ordinary consciousness as relatively unitary state. 

Consciousness is a brain function with both global and compartmental aspects. It succeeds in so far as it is a unitary state that integrates many different aspects of the brain-mind. Some of these are listed in Table 4. (See Figure 11 for physical orientation.)... 

The West Papua settlement was enshrined in the Papua of Indonesia Act of Autonomy of 2001, the preamble to which confirms that the integration of the nation must be maintained within the Unitary State of the Republic of Indonesia by respecting the equal and uniformed social and cultural life of the people of Papua through the formulation of a Special Autonomous region . Article 1 of the agreement then confirms that the Irian Jaya (West Papua) Province is granted special autonomy in the framework of the Unitary State of the Republic of Indonesia . The Article continnes ... 

Therefore the modulus of 4> and d is a measure for the total gap amplitude of the Cooper pairs at a given point k on the Fermi surface. In addition, the direction of the vector d specifies the relative contributions of the three triplet pair states. The above relation holds for triplet states which satisfy d x d = 0. They are called unitary states because they are invariant under time reversal. In this case the vector d defines a unique direction in spin space for every point on the Fermi surface. 

Do you accept the proposed special autonomy for East Timor within the Unitary State of the Republic of Indonesia. ACCEPT OR... 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

Although the two quantum models are defined somewhat dilferently, both QCA-I and QCA-II start with the same basic premise, endowing the classical system with two characteristically quantum features. They both (1) replace each site variable with a quantum state containing all fc classical site-color possibilities, and (2) introduce a quantum transition operator "I , defining mixed color —> mixed a)lor transitions. Only in QCA-II, however, is also unitary see discussion below. 

Having thus established at least a formal equivalency between a discretized field theory on a lattice and CA, Svozil invokes the so-called no-go theorem to show that field theory cannot be discretized in this simple fashion. The no-go theorem (see [karstSl] and [nielSl]) states essentially that under a set of only mild assumptions it is impossible to formulate a local, unitary, charge conserving lattice held theory without effectively doubling the size of the predicted fermion population (i.e. species doubling see discussion box). 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

The problem now is to find the corresponding Hamiltonian, t Hooft shows that the most obvious construction, obtained by rewriting U(t+l,t) as a product of cyclic elements, unfortunately does not work because at the end of the calculation there is no way to uniquely define the vacuum state. Given a cellular automaton with a local unitary evolution operator U = WgUg and the commutator [Ug, Ug ] 0 if [ af — af j> d for some d > 0, the real problem is therefore to find a Hamiltonian... 

Note that we are not transforming the vector operator itself—its form is independent of rotation. Hence, Eq. (7-11) must be satisfied by subjecting the states ijt to a unitary transformation U ... 

Among the usual advantages of such expressions as Eq. (7-80) and (7-81), one is salient they show forth the invariance of p and w with respect to the choice of the basis functions, u, in terms of which p, a, and P are expressed. The trace, as will be recalled, is invariant against unitary transformations, and the passage from one basis to another is performed by such transformations. The trace is also indifferent to an exchange of the two matrix factors, which is convenient in calculations. Finally, the statistical matrix lends itself to a certain generalization of states from pure cases to mixtures, required in quantum statistics and the theory of measurements we turn to this question in Section 7.9. 

Now let us use the set, <0> to form a matrix representation of some operator Q at time hi assuming that Q is not explicitly a function of time. The expectation value of Q in the various states, changes in time only by virtue of the time-dependence of the state vectors used in the representation. However, because this dependence is equivalent to a unitary transformation, the matrix at time t is derived from the matrix at time t0 by such a unitary transformation, and we know that this cannot change the trace of the matrix. Thus if Q — WXR our result entails that it is not possible to change the ensemble average of R, which is just the trace of Q. 

If the hamiltonians H(0) and H0(0) are such that there exist no bound states, and the states F )+ are properly normalized, l(+) is a unitary operator. Furthermore, it has the property that... 

The representation of these commutation rules is again fixed by the requirement that there exist no-particle states 0>out and 0>ln. The -matrix is defined as the unitary operator which relates the in and out fields ... 

Equation (11-165), which is the statement that the transition probabilities for bodily identical pairs of states be the same for all observers, is trivially satisfied for the Heisenberg-type description. On the other hand, for the Sehr6dinger-type description, Eq. (11-156) asserts that the one-to-one correspondence between the vectors Y> and T > is, in fact, either a unitary or anti-unitary 4 (norm preserving) one... 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

Representation theory for nonunitary groups.—Before proceeding we should consider what is meant by a unitary and an anti-unitary operator.5 -6 If the hamiltonian of a system commutes with the operators u and a of the group 0, and T and O are state functions of the system, u is unitary if... 

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