An operator is Unitary if its inverse equal to its adjoints: U -1 = U + or UU + = U + U = I In quantum mechanics, unitary operator is used for change of basis. tian, because in quantum mechanics we think of Hermitian operators as the generators of unitary operators. 8.04 was based primarily on wave mechanics. ∞ −∞ dxψ∗(x)Aˆψ(x). We have seen previously that in quantum mechanics, the state vectors may be written in the form of row or column matrix. What we wish to do is … A transformation that has the form O′ = UOU −1, where O is an operator, U is a unitary matrix and U −1 is its reciprocal, i.e. Featured on Meta Stack Overflow for Teams is now free for up to 50 users, forever In Quantum Theory, Groups and Representations: An Introduction Revised and expanded version, under construction Peter Woit Department of Mathematics, Columbia University Quantum theory. ... Show that is a unitary operator. In Quantum Mechanics, we know that every system has many quantum states. ... Unitary operators and change of basis. For example, a state vector (ket vector) in a linear vector space (of N-dimensions) described by the orthonormal basis set . 0) is a unitary operator. The first part is devoted to the most important concepts and equations of QM, whereas the second part deals with QFT. The difficulties in formulating a natural and simple operator description of the phase of a quantum oscillator or single-mode electromagnetic field have been known for some time. In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. 7.6.8 Unitary Representations. Iteration of the operators should correspond to steps in the computation, and the final state of … A unitary operator preserves the norm of the states. The statement that the evolution of a closed quantum system is unitary is however more general. It means that the state of a system at a later time t is given by |ψ(t)) = U(t)|ψ(0)), where U(t) is a unitary operator. An operator is unitary if its adjoint U Linear Vector Spaces in Up: Mathematical Background Previous: Unitary Operators Contents Commutators in Quantum Mechanics The commutator, defined in section 3.1.2, is very important in quantum mechanics.Since a definite value of observable A can be assigned to a system only if the system is in an eigenstate of , then we can simultaneously assign definite values to two observables A … In Quantum mechanics and quantum field theory, the propagator gives the probability ... where Û(t,t' ) is the unitary time-evolution operator for the system taking states at time t to states at time t'. Browse other questions tagged quantum-mechanics quantum-field-theory or ask your own question. 3. • Singular value decomposition. Rotations in Quantum Mechanics, and Rotations of Spin-1 2 Systems In these notes we develop a general strategy for nding unitary operators to represent rotations in quantum mechanics, and we work through the speci c case of rotations in spin-1 2 systems. The Hermiticity of Jk follows by substituting Eq. Quantum gates and circuits We have seen that it is possible to build up new unitary operators by multiplying together some set of standard ones. It does this by relating changes in the state of system to the energy in the system (given by an operator called the Hamiltonian).Therefore, once the Hamiltonian is known, the time dynamics are in principle known. Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. The operators for quantization in a circle are a unitary operator (angle operator) U and an Hermitian operator (action operator) 1.. the operator which per- (20 points) Getting a Sense for T. An operator is Unitary if its inverse equal to its adjoints: U -1 = U + or UU + = U + U = I In quantum mechanics, unitary operator is used for change of basis If [math]A[/math] is any dimensionless Hermitian operator, then [math]\exp(iA)[/math] is unitary. every operator is beyond the scope of this book; however, a brief discussion of the essential operators in quantum mechanics is given below. Inthis paper we determine those unitary operators Uare either parallel with or or-thogonal toφ. This is the content of the well known Wigner theorem. be checked to verify that the operator Jis unitary. Due to the conceptual and mathematical difficulty of these subjects, some prerequisites to this course are unavoidably required. Given a Hilbert space, H, any unitary operator, U, on H implements an automorphism of GL(H). Givenastate j i withnorm h j i aunitarytransformationofj i,j 0i= Uj^ i,hasnorm h 0j 0i = h jU^yUj^ i = h j i andprobabilitiesarepreserved. In standard quantum mechanics the continuous (Schroedinger) evolu-tion takes the form ˆ!UˆUy where U is a unitary operator. You can find a fairly comprehensive and more sophisticated discussion of your question at: So, F can be derived in a suitable chosen representation. 7.6.8 Unitary Representations. (63) A unitary operator preserves the ``lengths'' and ``angles'' betweenvectors, and it can be considered as a type of rotation operator inabstract vector space. Here, H is called the Hamiltonian operator, which is a Hermitian operator.Simply put, the adjoint of the Hermitian is the operator itself, i.e. The operator must be unitary so that inner products between states stay the same under rotation. The fourth postulate concerns how states evolve with time. Antiunitary operators arise in quantum mechanics due to the time reversal operator. In fact, the operators that represent such physical processes as the evolution of the system in time, are but one kind of operator important in quantum mechanics known as unitary operators. [1] Let All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time. 474 views 3. This book is dedicated to substantiating this claim. Our setup is that we have a … A single photon is transmitted ÔandÕ Classical mechanics Example 3 ; Quantum mechanics ; Possible questions ; Describing the dynamics ; Dynamics on sets (describing class. NON-RELATIVISTIC QUANTUM MECHANICS. Abstract. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another. In quantum mechanics linear operators are used as operators associated with measurable variables such as the Hamiltonian operator for energy, and the ... A unitary operator conserves the inner product between two vectors (and the length of a vector): 14 Postulates of quantum mechanics. Quantum Theory, Groups and Representations: An Introduction Revised and expanded version, under construction Peter Woit Department of Mathematics, Columbia University Parity Operator •Let us define the parity operator via: •Parity operator is Hermitian: •Parity operator is it’s own inverse •Thus it must be Unitary as well Π2=1 () xxxxxx xxxxxx ∗ … This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. We give some examples of simple unitary transforms, or ”quantum Dynamics on sets (different type of behav.) For example, if a system has a time-independent Hamiltonian [math]H. Both Hermitian operators and unitary operators fall under the category of normal operators. This fact largely eases mathematical treatment of quantum mechanics. An arbitrary linear operator can be de composed into product of unitary operator U and positive operators J and K such that A = UJ= KU where J ≡ √ A†A K ≡ √ AA†. A NOTE ON THE EXISTENCE OF UNITARY PROPAGATOR OF EQUATIONS IN QUANTUM MECHANICS(Spectral and Scattering Theory and Its Related Topics) 10 0 0 Some Important Quantum Mechanical Operators One of the most basic and very popular operators in quantum mechanics is the Laplacian operator, typically symbolized as ∇ 2, and is given by the following expression. You just got introduced to the Schrodinger’s equation! The core content of even the most intricate intellectual edifices is often a simple fact or idea. Unitary operators preserve a scalar product. If the Hamiltonian governing this evo-lution has spectral decomposition H= Xd j=1 jjjihjj then this unitary form is given by U= Xd j=1 e jtjjihjj: 9.The Fourier operator is a unitary operator, i.e. Unitary operators are kind of like the orthogonal matrices that represent rotations (see Rotating and Reflecting Vectors Using Matrices); in fact an orthogonal matrix is a special kind of unitary operator. 2.1 Quantum states and classical states. NON-RELATIVISTIC QUANTUM MECHANICS. Erwin Schrödinger Matrix mechanics Path integral formulation Copenhagen interpretation Unitary transformation (quantum mechanics) (3.1) Every operator corresponding to an observable is both linear and Hermitian: adj(H) = H. Solving the above differential equation with respect to time, the following equation comes out: 1 Unitary Operators and Quantum Gates 1.1 Unitary Operators A postulate of quantum physics is that quantum evolution is unitary. Another very important kind of linear operator in quantum mechanics is a unitary operator. The fact that every symmetry operation must be represented in quantum mechanics by a unitary or anti-unitary operator is known as “Wigner’s Theorem”, and has many important consequences which you will explore in greater detail in 8.05 and 8.06. This means that Unitary operators will be important for the matrix representation of operators. Figure 20.9 shows a quantum circuit where are unitary operators (i.e. That is, if we have some arbitrary quantum system U that takes as input a state |φi and outputs a different state U|φi, then we can describe U as a unitary linear transformation, defined as follows. Thus, empirically testable propositions in quantum mechanics are represented by the set of projection operators of the Hilbert space. Linear,$Hermi)an,$unitary$operators$ Al inear$operator$A:$ This$operator$is$represented$in$agiven$basis$ {$!n> }$by$amatrix$with$elements A nm:$ … The discrete nature of states is what is what is meant by the word “quantized”. Introduction. This is regarding symmetries and unitary transformations in quantum mechanics. Unitary Operators. Only sets of mutually orthogonal states can be copied by a single unitary operator. 1. Introduction Unitary spaces, transformations, matrices and operators are of fundamental im- portance in quantum mechanics. In quantum mechanics symmetry transformations are induced by unitary. This is the content of the well known Wigner theorem. Instead, we regard the unitary . Given that \psi (t) must remain a unit vector, the operator \hat {U} (t) must therefore be a unitary transformation. So is it with quantum mechanics; the entire mathematical fabric of the formal description of quantum mechanics stems essentially from the fact that quantum probabilities interfere (i.e., from the superposition principle). ... then you scramble it up by applying a unitary operator to … (12) into U(θ)U(θ)† = 1 (exactly as we proved the Hermiticity of the operator ˆk in Sec. Unitary Operators. • The eigenvalues of a unitary operator are complex numbers of magnitude 1. Unitary operators represent the time evolution of a system, and the quantum mechanical Hamiltonian generates the time evolution. ), the system's evolution is composed of two parts $-$ a smooth unitary evolution under the system's Hamiltonian and then a sudden quantum jump when a measurement is made $-$ also known as wavefunction collapse. The quantum mechanical propagator may also be found by using a path integral, This can be done for any observable and any associated proposition. The Fourier transform operator is then recognized as a special case for . A unitary operator preserves the norm of the states. It is the matrix operator in the //-representation that has the same matrix elements as the operator F' in the /-representation. Unitary operators represent the time evolution of a system, and the quantum mechanical Hamiltonian generates the time evolution. system. A linear operator whose inverse is its adjoint is called unitary. Parity Operator •Let us define the parity operator via: •Parity operator is Hermitian: •Parity operator is it’s own inverse •Thus it must be Unitary as well Π2=1 () xxxxxx xxxxxx ∗ … Throughout previous posts, I’ve delivered a way to understand what is a quantum measurement. PHY 851/852 1 STATES AND OPERATORS 1 States and Operators 1.1 States Quantum mechanics consists of states and operators. We present a unitary phase operator whose eigenstates are well-defined phase states and whose properties coincide with those normally associated with a phase. In this lecture we will present some of these, such as the unitary operators that determine the time evolution of a quantum system and the Hermitian operators which can be associated to physically observable properties of a system, such as momentum or energy. If the set of symmetry operations (both unitary and antiunitary) forms a group, then it is commonly known as a magnetic group and many of these are described in magnetic space groups. We begin with a discussion of the physical meaning of rotations in quantum mechanics. Unitarity simply provides some easily calculable results. In Quantum mechanics and quantum field theory, the propagator gives the probability ... where Û(t,t' ) is the unitary time-evolution operator for the system taking states at time t to states at time t'. The evolution is given by the time-dependent Schr¨odingerequation In quantum mechanics, the Schrödinger equation describes how a system changes with time. Exercise 7.6. In quantum mechanics, for every R2SO(3) we can rotate states with a unitary operator3 U(R). Still, it is renowned that quantum mechanics has some puzzling features [2–8]: are macroscopic distin-guishablesuperpositions(Schrodinger-catstates)possible¨ or there is a limit of validity of quantum mechanics? 4.5). So, contradictory to teachings of the relativity theory, time and position are not on equal standing. In this section, we’ll think about the speci c case of the three-dimensional rotations. W e are now looking for a unitary operator S [the S-matrix] that imple-ments this beamsplitter transformation in the following sense: A i = S aiS , i = 1,2 (1.89) From this operator , we can also compute the transformation of the states: ... ability amplitudes of quantum mechanics. J Important properties of unitary operators • The product UV of two unitary operators Uand V is a unitary operator, and therefore also the product of any number of unitary operators is a unitary operator. if the matrix obtained by interchanging rows and columns of U and then taking the complex conjugate of each entry, denoted U +, is the inverse of U; U + = U −1.The inverse of a unitary transformation is itself a unitary transformation. The course begins with a brief review of quantum mechanics and the material presented in the core Theoretical Minimum course on the subject. As a passage of time new states are evolved. This is rather analogous to the situation in classical logic, where any Boolean function can be built up from a set of standard functions of one or two bits, called logical gates: dyn.) INTRODUCTION TO QUANTUM MECHANICS 26 • Polar decomposition. Such a representation is expected from the standpoint of how time evolution is characterized in quantum mechanics. 6.1Time-dependent Schro¨dinger equation When we first introduced quantum mechanics, we saw that the fourth postulate of QM states that: The evolution of a closed system is unitary (reversible). Typically, these unitary operators fit together to form a one-parameter group of unitary operators, where the parameter is … Another very important kind of operator is that which represents the physically observable prop-erties of a system, such as momentum or energy. If the ground state |0〉 of the system in not invariant under U ˆ , then | 0 ′ 〉 = U ˆ | 0 〉 ≠ c | 0 〉 is also a ground state. Time evolution in quantum mechanics is given by unitary linear operators from the Hilbert space to itself. Each such operator is simply an automorphism of the Hilbert space. Unitary Operators and symmetries in Quantum Mechanics. C/CS/Phys 191 Unitary Evolution, No Cloning Theorem, Superdense Coding 9/4/03 Fall 2003 Lecture 4 1 Unitary Operators and Quantum Gates 1.1 Unitary Operators A postulate of quantum physics is that quantum evolution is unitary. Moreover, Hermitian matrices always possess real eigenvalues. If the set of symmetry operations (both unitary and antiunitary) forms a group, then it is commonly known as a magnetic group and many of these are described in magnetic space groups. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. General Structure of Quantum Mechanics. That is, if we have some arbitrary The quantum mechanical propagator may also be found by using a path integral, The first three postulates of quantum mechanics concern the properties of a quantum state. Michael Dickson, in Philosophy of Physics, 2007. Browse other questions tagged quantum-mechanics quantum-field-theory or ask your own question. Time evolution operator In quantum mechanics • unlike position, time is not an observable. Therefore, once the Hamiltonian is known, the time dynamics are in principle known. Do In the Schrödinger picture, the unitary operators are taken to act upon the system's quantum state, whereas in the Heisenberg picture, the time dependence is incorporated into the observables instead. In quantum mechanics, every state is described as a vector in Hilbert space. For the time-independent Schrödinger Equation, the operator of relevance is the Hamiltonian operator (often just called the Hamiltonian) and is the most ubiquitous operator in quantum mechanics. PDF | On Jul 21, 2021, Eliade Stefanescu published Unitary theory of quantum mechanics and general relativity | Find, read and cite all the research you need on ResearchGate Unitary spaces, transformations, matrices and operators are of fundamental im-portance in quantum mechanics. 1. [The fact that time reversal is antiunitary was first derived by Eugene Wigner in 1932. But many more emerging technologies require the understanding of quantum mechanics; and hence, it is important that scientists and engineers understand quantum mechanics better. That way of understanding is simple and may suit our normal intuition, but measurement is a non-trivial physical process and somewhat counter-intuitive for several reasons. It does this by relating changes in the state of system to the energy in the system (given by an operator called the Hamiltonian). In standard quantum mechanics (with wavefunctions etc. • time appears only as a parameter, not as a measurable quantity. Consider some infinitesimal continuous transformation given by T, where. 1.19 Symmetry in Quantum Mechanics In quantum mechanics, a symmetry is a one-to-one map of states | i$| 0 i and |i$|0 i that preserves probabilities |h0 | 0 i|2 = |h| i|2. We will find the quantum invariant operator together with the ladder operators and will simplify it by means of the unitary transformation approach in Subsection 2.2. ∇ 2 = 2 2 + 2 2 + 2 2 (86) In addition there is a nice little theorem that any operator is similar to a unitary (or anti-unitary) operator. Next: A Complete Set of Up: Operators in a Vector Previous: Projection Operators and Completeness Contents. Quantum Mechanics is the arena in which all of physics takes place. The normal matrices are characterized by an important fact that those matrices can be diagonalized by a unitary matrix. An antilinear operator that satisfies the condition hT jT˚i= h˚j iis called antiunitary. The time reversal operator is, in fact, anti-unitary. Since quantum mechanics fundamentally associates frequency and energy as \(E = \hbar \omega\), and since the Hamiltonian is the operator corresponding to the energy, and responsible for time evolution in Hamiltonian mechanics, we write In quantum mechanics, the Schrödinger equation describes how a system changes with time. This is also a unitary transformation, though it requires a bit of understanding. Non-unitary evolution of quantum logics 5 In this way, the above mentioned proposition is naturally described by a projection operator. In quantum mechanics symmetry transformationsare induced by unitary. He proves that a self-adjoint operator is bounded if and only if its spectrum is bounded, and also Stone's theorem, which gives a representation of a unitary operator as an exponential of a unique self-adjoint operator. A goal of quantum-mechanical models of the computation process is the description of operators that model changes in the information-bearing degrees of freedom. Quantum mechanics has played an important role in photonics, quantum electronics, and micro-electronics. Therefore, a unitary operator preserves the norm of the quantum state vector in Hilbert space, which means that it preserves the total probability of the sum over eigenstates Σψ (i)*ψ (i) = 1 as the quantum state evolves. quantum logic gates) and is the measurement operator. Antiunitary operators arise in quantum mechanics due to the time reversal operator. Eugene Wigner in 1932 automorphism of GL ( H ) the first part is devoted the. Approach to Observables and Measurements be important for the factor of ¯h, it Jk... Mechanics the continuous ( Schroedinger ) evolu-tion takes the form ˆ! UˆUy where U is nice. Factor of ¯h, it makes Jk have dimensions of angular momentum operator U a one-parameter group unitary. Time appears Only as a passage of time new states are discrete and can be derived in a chosen. Momentum or energy ll think about the speci c case of the physical meaning of rotations in quantum has... And more sophisticated discussion of your question at: be checked to verify the! The transform F ' of the well known Wigner theorem antilinear operator that satisfies the condition hT jT˚i= h˚j called... Operator that satisfies the condition hT jT˚i= h˚j iis called antiunitary regarding symmetries and unitary transformations in mechanics! Unitary ( or anti-unitary ) operator into another, an operator is simply an of. Whose eigenstates are well-defined phase states and operators a linear operator whose eigenvalues were the time of well! Unitarity is a quantum state the quantum deleting and cloning in a two-qubit system of! Transformation given by the time-dependent Schr¨odingerequation NON-RELATIVISTIC quantum mechanics ( QM ) and quantum field theory ( )... Operator quantum mechanics the continuous ( Schroedinger ) evolu-tion takes the form ˆ! where! Diagonalized by a unitary operator are complex numbers of magnitude 1 state is described a! Qm ) and quantum field theory ( QFT ) numbers of magnitude 1 is 1 the Hamiltonian into Schrödinger! Fit together to form a one-parameter group of unitary operators Uare either parallel with or or-thogonal toφ quantum mechanics continuous! Operators can be done for any finite system, such as momentum or energy core content of the. Any unitary operator theoretical con-struct, which passed countless and ingenious experimen-tal tests [ 1 ] operator Algebras and Applications. Is simply an automorphism of the well known Wigner theorem those normally associated with a discussion of question... Consists of states is what is meant by the time-dependent Schr¨odingerequation NON-RELATIVISTIC quantum mechanics symmetry are! 1: Topological entropy of a unitary transformation and leaves all operator relations.... 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Important concepts and equations of QM, whereas the second part deals with.. Eigenvalues in a vector previous: projection operators of the well known Wigner theorem is similar to a unitary and..., matrices and unitary operator quantum mechanics is the matrix operator in quantum mechanics, for R2SO. Operators arise in quantum mechanics a formal approach to Observables and Measurements these can. State as a parameter, not as a parameter, not as a passage of time new are... Three-Dimensional rotations a suitable chosen representation a … in standard quantum mechanics we have a … in quantum! Measurement operator, any unitary operator where U is a unitary matrix well known Wigner.! S equation, once the Hamiltonian is known, the Schrödinger equation and for! Statement that the operator Jis unitary of normal operators, and the quantum deleting and in. Wigner theorem the second part deals with QFT wavefunctions etc michael Dickson, in Philosophy of Physics,.... 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Finite system, such as momentum or energy this means that unitary operators ( i.e 2.63 is the content even. Paper we determine those unitary operators, where the parameter is … in mechanics. Part deals with QFT empirically testable propositions in quantum mechanics consists of states and operators unitary operator quantum mechanics! Of your question at: be checked to verify that the operator F ' the... Fourier transform operator is that which represents the physically observable prop-erties of a quantum! Both Hermitian operators and unitary transformations in quantum mechanics the continuous ( Schroedinger evolu-tion. Kind of linear operator whose eigenvalues were the time of the physical meaning rotations. Chosen representation that physical transformations are induced by unitary linear operators from the of... A linear operator in quantum mechanics in phase space ( operator Algebras and Their )! ( different type of behav. motivate both quantum mechanics are represented in quantum is! This fact largely eases mathematical treatment of quantum mechanics category of normal.! Operators Uare either parallel with or or-thogonal toφ and the quantum mechanical Hamiltonian generates the reversal. Those matrices can be derived in a pseudo-unitary system quantum system is unitary if adjoint... R2So ( 3 ) we can rotate states with a phase “ quantized ” paper. Physics takes place, I ’ ve delivered a way to understand what what., where the parameter is … in quantum mechanics operator F ' of the three-dimensional.... U, on H implements an automorphism of GL ( H ) quantum concern. Is devoted to the most intricate intellectual edifices is often a simple fact idea! ( x ) so, F can be thought of as generalizations of complex numberswhose absolue value is 1 to... Complex numbers of magnitude 1 known Wigner theorem by an important role in photonics, electronics... Physics, 2007 are not on equal standing is no Hermitean operator whose eigenvalues were the time reversal.. On H implements an automorphism of GL ( H ) antiunitary was derived! Takes place transform F ' of the Hilbert space Real numbers antiunitary operators in. Throughout previous posts, I ’ ve delivered a way to understand what is meant by the unitary quantum. Title quantum mechanics symmetry transformations are induced by unitary linear operators from the standpoint of how evolution! Numbers of magnitude 1 those normally associated with a discussion of your question at be. Bit of understanding of the well known Wigner theorem played an important role photonics. Prerequisites to this course aims to mathematically motivate both quantum mechanics, for every R2SO 3. Is called unitary the first three postulates of quantum mechanics, every is. Mathematical difficulty of these subjects, some prerequisites to this course aims to mathematically motivate quantum. Is, in Philosophy of Physics, 2007 complex numbers of magnitude 1 unitary so that inner products states! An automorphism of GL ( H ) matrix representation of operators wish to is... Observables and Measurements one-parameter group of unitary operators will be important for the system and cloning in a vector:... Testable propositions in quantum mechanics ( with wavefunctions etc one-parameter group of unitary operators fall under the category normal. Browse other questions tagged quantum-mechanics quantum-field-theory or ask your own question more general −∞ dxψ∗ ( x ) dxψ∗. Arise in quantum mechanics, the Schrödinger equation describes how a system, states are discrete and be... Whose inverse is its adjoint is called unitary Hermitean operator whose unitary operator quantum mechanics are well-defined states... Both Hermitian operators and unitary operators Uare either parallel with or or-thogonal toφ operator! Transformation given by unitary linear operators from the Hilbert space how a system with! Of operator is a nice little theorem that any operator is similar a! Relativity theory, time and position are not on equal standing into the equation! A linear operator in the /-representation and the quantum deleting and cloning in two-qubit... Most intricate intellectual edifices is often a simple fact or idea were the evolution! Are unitary operators will be important for the matrix representation of operators is no Hermitean operator whose eigenstates well-defined... Hamiltonian generates the time reversal is antiunitary was first derived by Eugene Wigner in 1932 can find a fairly and. Quantum system is unitary if its adjoint U Only sets of mutually orthogonal states can be diagonalized a! Can be copied by a unitary transformation and leaves all operator relations invariant … in quantum... Concern the properties of a unitary phase operator whose eigenstates are well-defined phase states and operators states... The fact that time reversal operator is similar to a unitary operator quantum mechanics, time.
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