< how to keep a worm hole functional and kept stable

> The universe decaying to a new vacuum state[edit] \, , Jump up ^ E. C. G. Sudarshan "Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams", Phys. Rev. Lett.,10 (1963) pp. 277–279. doi:10.1103/PhysRevLett.10.277  Ket state	constant	 | \psi_{I}(t) \rang = e^{i H_{0, S} ~t / \hbar} | \psi_{S}(t) \rang 	 | \psi_{S}(t) \rang = e^{-i H_{ S} ~t / \hbar} | \psi_{S}(0) \rang  so that the matrix L is constant in time: it is conserved. 
< 
>  To do this, he investigated the action integral as a matrix quantity, Property or effect	Nomenclature	Equation  If the action contains terms which multiply ? and x, at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism. 
< 
>      . One particle	N particles There are several problems with this integral, all stemming from the incompatibility of the matrix formalism with the old picture of orbits. Which period T should be used? Semiclassically, it should be either m or n, but the difference is order h, and an answer to order h is sought. The quantum condition tells us that Jmn is 2pn on the diagonal, so the fact that J is classically constant tells us that the off-diagonal elements are zero. 
< 
> The many-worlds interpretation is very vague about the ways to determine when splitting happens, and nowadays usually the criterion is that the two branches have decohered. However, present day understanding of decoherence does not allow a completely precise, self-contained way to say when the two branches have decohered/"do not interact", and hence many-worlds interpretation remains arbitrary. This objection is saying that it is not clear what is precisely meant by branching, and point to the lack of self-contained criteria specifying branching. \mathbf{F} = m \mathbf{a} \, . Classical mechanics was traditionally divided into three main branches: 3.2 Free particle \, . 
< 
>  \, . This article is about a class of mechanics theories. For hidden variables in economics, see latent variable. For other uses, see Hidden variables (disambiguation). All the forms of the equation of motion above say the same thing, that A(t) is equivalent to A(0), through a basis rotation by the unitary matrix eHt, a systematic picture elucidated by Dirac in his bra–ket notation. 
< 
>    | \psi_i\rangle \langle \psi_i| \,  i\hbar\frac{d}{dt}A_I(t)=\left[A_I(t),H_0\right].\; 4.4 As a probability   The Dirac theory, while providing a wealth of information that is accurately confirmed by experiments, nevertheless introduces a new physical paradigm that appears at first difficult to interpret and even paradoxical. Some of these issues of interpretation must be regarded as open questions.[citation needed] Frisch, R.; Stern, O. (1933). "Über die magnetische Ablenkung von Wasserstoffmolekülen und das magnetische Moment des Protons. I". Zeitschrift für Physik 85: 4. Bibcode:1933ZPhy...85....4F. doi:10.1007/BF01330773. 
< 
>  Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill. \{a, b\} = ab + ba  David Wallace, Harvey R. Brown, Solving the measurement problem: de Broglie–Bohm loses out to Everett, Foundations of Physics, arXiv:quant-ph/0403094  where 
< 
>  Jump up ^ A response to Bryce DeWitt[dead link], Martin Gardner, May 2002 This is the original form of Heisenberg's equation of motion. Equations[show]   P_\mathrm{op}\psi = mc\psi. \, Jump up ^ Georgescu, G. and C. Vraciu. 1970, On the characterization of centered Lukasiewicz algebras., J. Algebra, 16: 486–495. where the bracket expression 
< 
>  For the perturbation Hamiltonian H1,I, however, K(x,y;T) = \int_{x(0)=x}^{x(T)=y} \Pi_t \exp\left\{-{1\over 2} \left({x(t+\epsilon) -x(t) \over \epsilon}\right)^2 \epsilon \right\} Dx In special relativity, the momentum of a particle is given by Main article: quantum harmonic oscillator ? = position-space wavefunction \left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t\right)\psi = \kappa\psi Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group U(t): H ? H such that 6 Non-relativistic dynamics  
< 
> James P. Hogan, The Proteus Operation (science fiction involving the many-worlds interpretation, time travel and World War 2 history), Baen, Reissue edition (August 1, 1996) ISBN 0-671-87757-7 Interpretations[show]   MWI response: the splitting can be regarded as causal, local and relativistic, spreading at, or below, the speed of light (e.g., we are not split by Schrödinger's cat until we look in the box).[68][unreliable source?] For spacelike separated splitting you can't say which occurred first — but this is true of all spacelike separated events, simultaneity is not defined for them. Splitting is no exception; many-worlds is a local theory.[49] Ahmad, Ishfaq (1971). Mathematical Integrals in Quantum Nature. The Nucleus. pp. 189–209.  (f \star g)(x,p) = \frac{1}{\pi^2 \hbar^2} \, \int f(x+x',p+p') \, g(x+x'',p+p'') \, \exp{\left(\tfrac{2i}{\hbar}(x'p''-x''p')\right)} \, dx' dp' dx'' dp'' ~. 
< 
> then ?t obeys the free Schrödinger equation just as K does: The need for regulators and renormalization[edit] Main article: Ehrenfest Theorem By setting n_1 to 1 and letting n_2 run from 2 to infinity, the spectral lines known as the Lyman series converging to 91 nm are obtained, in the same manner:   
< 
> In the correspondence limit, when indices m, n are large and nearby, while k,r are small, the rate of change of the matrix elements in the diagonal direction is the matrix element of the J derivative of the corresponding classical quantity. So its possible to shift any matrix element diagonally through the correspondence,  See also: Alternate History Dirac spinor  A[x]=\exp\left(\frac{1}{\hbar}\int X(t)\,\mathrm{d}t\right) \text{.} The above two equations are the Ward–Takahashi identities. The ?-genvalue equation for the static Wigner function then reads 6 Non-relativistic dynamics 
< 
>   \dot q = {\partial H \over \partial p} \,  Jump up ^ Herbert S. Green (1965), "Matrix mechanics" (P. Noordhoff Ltd, Groningen, Netherlands) ASIN : B0006BMIP8. s = spin quantum number \frac{\partial\psi}{\partial t} = {\rm i}\cdot \left[ \frac{1}{2}\nabla^2  -  V(x)\right]\psi 
< 
>  Time-evolution equations in the interaction picture[edit] Wheeler–Feynman absorber theory  
< 
> |\psi\rangle = \int_x \psi(x)|x\rangle [P_i, P_j ] = 0 where H is the Hamiltonian and [•,•] denotes the commutator of two operators (in this case H and A). Taking expectation values automatically yields the Ehrenfest theorem, featured in the correspondence principle. Jump up ^ M. Born and P. Jordan, Zur Quantenmechanik, Zeitschrift für Physik, 34, 858-888, 1925 (received September 27, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1] Jump up ^ Breitbart.com, Parallel universes exist – study, Sept 23 2007  
< 
> [ ] is the commutator = \int_0^T dt \left( {dP\over dJ} {dX\over dt} + P{d\over dJ}{dX\over dt} \right) Density matrix	constant	\rho_I (t)=e^{i H_{0, S} ~t / \hbar}  \rho_S (t) e^{-i H_{0, S}~ t / \hbar}	\rho_S (t)=  e^{-i H_{ S} ~t / \hbar} \rho_S(0) e^{i H_{ S}~ t / \hbar}  Newtonian mechanics  (\gamma^\mu)^\dagger\gamma^0 = \gamma^0\gamma^\mu \,, Two classes of particles with very different behaviour are bosons which have integer spin (S = 0, 1, 2...), and fermions possessing half-integer spin (S = 1/2, 3/2, 5/2, ...). which implies Jeffrey A. Barrett and Peter Byrne, eds., "The Everett Interpretation of Quantum Mechanics: Collected Works 1955–1980 with Commentary", Princeton University Press, 2012. Henceforth we identify elements of Q with self-adjoint projections on the Hilbert space H. 
< 
> Quantum tunneling[edit] Pure states[edit] Textbooks[edit] Dirac, P. A. M. (1930). "A Theory of Electrons and Protons". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 126 (801): 360. Bibcode:1930RSPSA.126..360D. doi:10.1098/rspa.1930.0013. JSTOR 95359. Schrödinger equation   \Rightarrow - \Delta E_\mathrm{p} = \Delta E_\mathrm{k} \Rightarrow \Delta (E_\mathrm{k} + E_\mathrm{p}) = 0 \, .  
< 
> power	kg·m2·s-3   \Psi = \psi(\mathbf{r}) e^{-iEt/\hbar} 	 \Psi = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N)  Jump up ^ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0 Before many-worlds, reality had always been viewed as a single unfolding history. Many-worlds, however, views reality as a many-branched tree, wherein every possible quantum outcome is realised.[12] Many-worlds reconciles the observation of non-deterministic events, such as the random radioactive decay, with the fully deterministic equations of quantum physics.  
< 
> MWI response: The splitting is time asymmetric; this observed temporal asymmetry is due to the boundary conditions imposed by the Big Bang[59]   Thaller, B. (1992). The Dirac Equation. Texts and Monographs in Physics. Springer. H_S = H_{0,S} + H_{1, S} ~.  Jump up ^ Lutz Polley, Position eigenstates and the statistical axiom of quantum mechanics, contribution to conference Foundations of Probability and Physics, Vaxjo, Nov 27 – Dec 1, 2000 R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999. An extraordinarily lucid discussion of some logical and philosophical issues of quantum mechanics, with careful attention to the history of the subject. Also discusses consistent histories. 
< 
>    Quantum logic has some properties that clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:[6] rotating each state |x\rangle by a phase f(x), that is, redefining the phase of the wavefunction:  ^ Jump up to: a b c Bryce Seligman DeWitt, The Many-Universes Interpretation of Quantum Mechanics, Proceedings of the International School of Physics "Enrico Fermi" Course IL: Foundations of Quantum Mechanics, Academic Press (1972)  11 References 
< 
> where, as always, there is an implied summation over the twice-repeated index k = 1, 2, 3. This looks promising, because we see by inspection the rest energy of the particle and, in case A = 0, the energy of a charge placed in an electric potential qA0. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is Jump up ^ "The Many Minds Approach". 25 October 2010. Retrieved 7 December 2010. This idea was first proposed by Austrian mathematician Hans Moravec in 1987... Interaction picture The path integral formulation was very important for the development of quantum field theory. Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time, and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation   \lang A \rang _t = \lang \psi (t) | A | \psi(t) \rang.  
< 
> Jump up ^ Constance Ried Courant (Springer, 1996) p. 93. Where W(\Tau) is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor, and can be absorbed into the constant a. \int \mathcal{D}\phi\, Q\left[F e^{iS}\right][\phi]=0,  i\hbar\frac{d}{dt}A_I(t)=\left[A_I(t),H_0\right].\;  
< 
> Continuum mechanics, for materials modelled as a continuum, e.g., solids and fluids (i.e., liquids and gases). (i\gamma^\mu\partial_\mu^\prime - m)\psi^\prime(x^\prime,t^\prime) = 0.  Further, different choices of canonical variables lead to very different seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons, the Feynman path integral has made earlier formalisms largely obsolete. \,.   
< 
> ? = 2pf is the angular frequency and frequency of the particle Jump up ^ M. S. Marinov, A new type of phase-space path integral, Phys. Lett. A 153, 5 (1991). Using {\partial\!\!\!\big /} (pronounced: "d-slash"[4]) in Feynman slash notation, which includes the gamma matrices as well as a summation over the spinor components in the derivative itself, the Dirac equation becomes:   
< 
>  after the measurement, the system now will be in the state Statistical structure[edit] Garden of Forking Paths U^{-1} H U = H where the symbols p, q and r are propositional variables. To illustrate why the distributive law fails, consider a particle moving on a line and let \langle \, \rangle denotes the average 3 Probability Griffiths, D.J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2. Max Born The statistical interpretation of quantum mechanics. Nobel Lecture – December 11, 1954. 
< 
> Note that it follows easily from uniqueness from Kadison's theorem that  His crucial insight was to differentiate the quantum condition with respect to n. This idea only makes complete sense in the classical limit, where n is not an integer but the continuous action variable J, but Heisenberg performed analogous manipulations with matrices, where the intermediate expressions are sometimes discrete differences and sometimes derivatives. Heisenberg's uncertainty principles	 
< 
> \int \psi_0(x) \int_{u(0)=x} -\left( \int \left({d\over dt} {\partial S\over \partial \dot{u}} - {\partial S \over \partial u}\right)\epsilon(t) dt \right) e^{iS} Du In his 1957 doctoral dissertation, Everett proposed that rather than modeling an isolated quantum system subject to external observation, one could mathematically model an object as well as its observers as purely physical systems within the mathematical framework developed by Paul Dirac, von Neumann and others, discarding altogether the ad hoc mechanism of wave function collapse. Since Everett's original work, there have appeared a number of similar formalisms in the literature. One such idea is discussed in the next section. {dx \over dt} = {x(t+\epsilon) - x(t) \over \epsilon}  The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities fail. 
< 
> Observable	A_H (t)=e^{i H_{ S}~ t / \hbar} A_S e^{-i H_{ S}~ t / \hbar}	A_I (t)=e^{i H_{0, S} ~t / \hbar} A_S e^{-i H_{0, S}~ t / \hbar} 	constant  MWI response: Everett analysed branching using what we now call the "measurement basis". It is fundamental theorem of quantum theory that nothing measurable or empirical is changed by adopting a different basis. Everett was therefore free to choose whatever basis he liked. The measurement basis was simply the simplest basis in which to analyse the measurement process.[63][64] Jump up ^ Wojciech Hubert Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics, 75, pp 715–775, (2003)  In Bohm's interpretation, the (nonlocal) quantum potential constitutes an implicate (hidden) order which organizes a particle, and which may itself be the result of yet a further implicate order: a superimplicate order which organizes a field.[17] Nowadays Bohm's theory is considered to be one of many interpretations of quantum mechanics which give a realist interpretation, and not merely a positivistic one, to quantum-mechanical calculations. Some consider it the simplest theory to explain quantum phenomena.[18] Nevertheless it is a hidden variable theory, and necessarily so.[19] The major reference for Bohm's theory today is his book with Basil Hiley, published posthumously.[20] Further reading[edit]  Velocity and speed[edit]  
< 
>    and the partial derivative now is with respect to p at fixed q.  MWI response: the decoherence or "splitting" or "branching" is complete when the measurement is complete. In Dirac notation a measurement is complete when: H = {1\over 2} P^2 + {1\over 2} X^2  + \epsilon X^3 ~. Jump up ^ M.O.Scully and H.Walther, Phys.Rev. A39,5229 (1989). 
< 
> T L dt, which is just the action function which classical mechanics requires to be stationary for small variations in all the intermediate qs. This shows the way in which equation (11) goes over into classical results when h becomes extremely small." Thus pure states can be identified with rays in the Hilbert space H.  Branches[edit]  \gamma^2 = \left(\begin{array}{cccc} 0 & \sigma_y \\ -\sigma_y & 0 \end{array}\right), 
< 
>  then we have the transformed Dirac equation in a way that demonstrates manifest relativistic invariance:  Jump up ^ P.W. Atkins (1974). Quanta: A handbook of concepts. Oxford University Press. p. 52. ISBN 0-19-855493-1. 
< 
> Jump up ^ A.Albrecht, Phys.Rev. D48,3768 (1993).  \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right). Lagrangian mechanics \, E = c\sqrt{p^2 + m^2c^2}\,, 
< 
> In the language of functional analysis, we can write the Euler–Lagrange equations as 5. It is not clear how the transactional interpretation handles the quantum mechanics of more than one particle. This equation is referred to as the Schwinger–Tomonaga equation.  p = hf/c = h/\lambda\,\! \sigma_x \sigma_p \ge \frac{\hbar}{2} 4.5 Schwinger–Dyson equations In any representation of the phase space distribution with its associated star product, this is 
< 
> So the quantum condition integral is the average value over one cycle of the Poisson bracket of X and P. MacKenzie, Richard (2000). "Path Integral Methods and Applications". arXiv:quant-ph/0004090. Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill.   The new elements in this equation are the 4 × 4 matrices ak and ß, and the four-component wave function ?. There are four components in ? because evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron (see below for further discussion). 
< 
>  Jump up ^ M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik II, Zeitschrift für Physik, 35, 557-615, 1925 (received November 16, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1] Non-relativistic time-dependent Schrödinger equation[edit] Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill. 
< 
> See also[edit]     \lambda_{\mathrm{vac}} \! is the wavelength of electromagnetic radiation emitted in vacuum, 
< 
>  Since the frequencies which the particle emits are the same as the frequencies in the fourier description of its motion, this suggests that something in the time-dependent description of the particle is oscillating with frequency (En-Em)/h. Heisenberg called this quantity Xnm, and demanded that it should reduce to the classical Fourier coefficients in the classical limit. For large values of n, m but with n - m relatively small, Xnm is the (n-m)th Fourier coefficient of the classical motion at orbit n. Since Xnm has opposite frequency to Xmn, the condition that X is real becomes \mathbf{a} = {\mathrm{d}\mathbf{v} \over \mathrm{d}t} = {\mathrm{d^2}\mathbf{r} \over \mathrm{d}t^2}. Wavefunction probability density	?	\rho = \left | \Psi \right |^2 = \Psi^* \Psi 	m-3	[L]-3 
< 
> \sigma(A)\sigma(B) \geq \frac{1}{2}\langle i[\hat{A}, \hat{B}] \rangle J = total 2m K_\mathrm{NR}(p) = {i \over (p_0-m) - {\vec{p}^2\over 2m} } U(t)=\mathcal{T}\left[\exp\left(-\frac{i}{\hbar} \int_{t_0}^t \,{\rm d}t'\, H(t')\right)\right]\,, force	kg·m·s-2 Static forces and virtual-particle exchange 
< 
> t = time List of mathematical tools[edit] Alternatively, a division can be made by region of application: In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian H' =H0. If the multiplications implicit in this formula are reinterpreted as matrix multiplications, what does this mean? Einstein–Maxwell–Dirac equations \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+\frac{\partial A}{\partial t},  \dot q = {\partial H \over \partial p} \, 
< 
> List of mathematical tools[edit] HPO formalism (An approach to temporal quantum logic) Differentiating both equations once more and solving for them with proper initial conditions,   where, as always, there is an implied summation over the twice-repeated index k = 1, 2, 3. This looks promising, because we see by inspection the rest energy of the particle and, in case A = 0, the energy of a charge placed in an electric potential qA0. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is  Breit equation 
< 
> MWI response: the decoherence or "splitting" or "branching" is complete when the measurement is complete. In Dirac notation a measurement is complete when: \partial_\mu \left( \bar{\psi}\gamma^\mu\psi \right) = 0. Time evolution from t0 to t may be viewed as a two-step time evolution, first from t0 to an intermediate time t1, and then from t1 to the final time t. Therefore, Interpretations of quantum mechanics (i\gamma^\mu\partial_\mu^\prime - m)\psi^\prime(x^\prime,t^\prime) = 0. Then, from the properties of the functional integrals 
< 
>    \operatorname{P}\!\left(\sum_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i).  While this restriction correctly selects orbits with more or less the right energy values En, the old quantum mechanical formalism did not describe time dependent processes, such as the emission or absorption of radiation. Statistical mechanics, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk thermodynamic properties of materials.  The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the Poynting vector divided by c2, where c is the speed of light in free space.  \Psi = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N)  
< 
> If the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with H1,S, leaving H0,S time-independent. We proceed assuming that this is the case. If there is a context in which it makes sense to have H0,S be time-dependent, then one can proceed by replacing e^{\pm i H_{0,S} t/\hbar} by the corresponding time-evolution operator in the definitions below. \, \tilde{K}(p;T) = \tilde{G}_\epsilon(p)^{T/\epsilon} ^ Jump up to: a b Osnaghi, Stefano; Freitas, Fabio; Olival Freire, Jr (2009). "The Origin of the Everettian Heresy" (PDF). Studies in History and Philosophy of Modern Physics 40: 97–123. doi:10.1016/j.shpsb.2008.10.002. then ?t obeys the free Schrödinger equation just as K does: Since this formulation of quantum mechanics is analogous to classical action principles, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case. 
< 
> Offer waves (OW) obey the Schrödinger equation and confirmation waves (CW) obey the complex conjugate Schrödinger equation. A transaction is a genuinely stochastic event, and therefore does not obey a deterministic equation. Outcomes based on actualized transactions obey the Born rule and, as noted in Cramer (1986), TI provides a derivation of the Born rule rather than assuming it as in standard quantum mechanics (QM).  Dispersion of observable	  X_n = X_{-n}^*  .  Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation. The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function (see below). The representation shown here is known as the standard representation - in it, the wave function's upper two components go over into Pauli's 2-spinor wave function in the limit of low energies and small velocities in comparison to light. Jump up ^ Joseph Gerver, The past as backward movies of the future, Physics Today, letters followup, 24(4), (April 1971), pp 46–7  
< 
> This is called the propagator. Superposing different values of the initial position x with an arbitrary initial state \psi_0(x) constructs the final state. Star product[edit] The operator identity Jump up ^ Wojciech Hubert Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics, 75, pp 715–775, (2003)  For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near p0 = m. When convolving with the propagator, which in p space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near p0 = m, the dominant first term has the form: 
< 
> Mathematical development[edit] The ray approximation of classical mechanics breaks down when the de Broglie wavelength is not much smaller than other dimensions of the system. For non-relativistic particles, this wavelength is Consistent histories  
< 
>  Quantum logic can be formulated either as a modified version of propositional logic or as a noncommutative and non-associative many-valued (MV) logic.[1][2][3][4][5] From this, the commutator of Lz and the coordinate matrices X, Y, Z can be read off, ^ Jump up to: a b Everett [1956]1973, "Theory of the Universal Wavefunction", chapter 6 (e) E. Prugovecki, Quantum Mechanics in Hilbert Space, Dover, 1981.  
< 
>  This is called the Ito lemma in stochastic calculus, and the (euclideanized) canonical commutation relations in physics. Jump up ^ "Most of the Good Stuff", Memories Of Richard Feynman, edited by Laurie M. Brown and John S. Rigden, American Institute of Physics, the chapter by Murray Gell-Mann. µ = chemical potential which is precisely the partition function of statistical mechanics for the same system at temperature quoted earlier. One aspect of this equivalence was also known to Schrödinger who remarked that the equation named after him looked like the diffusion equation after Wick rotation.  This factor is needed to restore unitarity. 
< 
> Forces; Newton's second law[edit]  Contents  [hide]  Marchildon, L. (2006). “Causal Loops and Collapse in the Transactional Interpretation of Quantum Mechanics,” Physics Essays 19, 422. with the conservation of probability current and density following from the Schrödinger equation: Q is also sequentially complete: any pairwise disjoint sequence{Vi}i of elements of Q has a least upper bound. Here disjointness of W1 and W2 means W2 is a subspace of W1?. The least upper bound of {Vi}i is the closed internal direct sum. Defining the time order to be the operator order:  
< 
> See also: Fermionic field Main article: Schwinger–Dyson equation  i\hbar\frac{d}{dt}A_I(t)=\left[A_I(t),H_0\right].\; Jump up ^ Simon Saunders: Derivation of the Born rule from operational assumptions. Proc. Roy. Soc. Lond. A460, 1771–1788 (2004). F' = F (the force on a particle is the same in any inertial reference frame) ^ Jump up to: a b Steven Weinberg Testing Quantum Mechanics, Annals of Physics Vol 194 #2 (1989), pg 336–386 
< 
> 12 Notes  E - e\phi \approx mc^2  \int\limits_{-\infty}^{+\infty}\,\ldots \int\limits_{-\infty}^{+\infty}\, 5 Automorphisms   
< 
> Angular momentum[edit] Spectrum	? = wavelength of emitted photon, during electronic transition from Ei to Ej	\frac{1}{\lambda} = R\left(\frac{1}{n_j^2} - \frac{1}{n_i^2}\right), \, n_j<n_i\,\! = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2} + V(x_1,x_2,\cdots x_N,t)   e^{i\mathcal{S}[\phi]}, so that the matrix L is constant in time: it is conserved.  E\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi 	 E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi   This principle of uncertainty holds for many other pairs of observables as well. For example, the energy does not commute with the position either, so it is impossible to precisely determine the position and energy of an electron in an atom. 
< 
> \frac{d}{dt}\hat{A}(t)=\frac{i}{\hbar}[\hat{H},\hat{A}(t)]+\frac{\partial \hat{A}(t)}{\partial t},  Equations of motion[edit] Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the canonical commutation relations from this rule. This was done by Feynman.[4] 
< 
> Tunneling is a hallmark quantum effect where a quantum particle, not having sufficient energy to fly above, still goes through a barrier. This effect does not exist in classical mechanics. Advanced topics[show] While the position, velocity and acceleration of a particle can be referred to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. An inertial frame is such that when an object without any force interactions (an idealized situation) is viewed from it, it appears either to be at rest or in a state of uniform motion in a straight line. This is the fundamental definition of an inertial frame. They are characterized by the requirement that all forces entering the observer's physical laws[clarification needed] originate in identifiable sources (charges, gravitational bodies, and so forth). A non-inertial reference frame is one accelerating with respect to an inertial one, and in such a non-inertial frame a particle is subject to acceleration by fictitious forces that enter the equations of motion solely as a result of its accelerated motion, and do not originate in identifiable sources. These fictitious forces are in addition to the real forces recognized in an inertial frame. A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that are unaccelerated with respect to the distant stars (an extremely distant point) are regarded as good approximations to inertial frames. U(t)=\mathcal{T}\left[\exp\left(-\frac{i}{\hbar} \int_{t_0}^t \,{\rm d}t'\, H(t')\right)\right]\,, In the Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. For time evolution from a state vector |\psi(t_0)\rangle at time t_0 to a state vector |\psi(t)\rangle at time t, the time-evolution operator is commonly written U(t, t_0), and one has  If the multiplications implicit in this formula are reinterpreted as matrix multiplications, what does this mean? H = \frac{1}{2m}\left(p  - \frac{e}{c}A\right)^2 + e\phi - \frac{e\hbar}{2mc}\sigma\cdot B. Highfield, Roger (September 21, 2007). "Parallel universe proof boosts time travel hopes". The Daily Telegraph. Archived from the original on 2007-10-20. Retrieved 2007-10-26.. 
< 
> surface tension	kg·s-2 \left\langle \frac{\delta F[\phi]}{\delta \phi} \right\rangle = -i \left\langle F[\phi]\frac{\delta \mathcal{S}[\phi]}{\delta\phi} \right\rangle   Some of these stories or films violate fundamental principles of causality and relativity, and are extremely misleading since the information-theoretic structure of the path space of multiple universes (that is information flow between different paths) is very likely extraordinarily complex. Also see Michael Clive Price's FAQ referenced in the external links section below where these issues (and other similar ones) are dealt with more decisively.  The Dirac theory, while providing a wealth of information that is accurately confirmed by experiments, nevertheless introduces a new physical paradigm that appears at first difficult to interpret and even paradoxical. Some of these issues of interpretation must be regarded as open questions.[citation needed] 
< 
> (i\gamma^\mu(\partial_\mu + ieA_\mu) - m) \psi = 0\, dX = i[X,P] ds = ds Historically, in physics, hidden variable theories were espoused by some physicists who argued that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system; i.e., that quantum mechanics is ultimately incomplete, and that a complete theory would provide descriptive categories to account for all observable behavior and thus avoid any indeterminism. The existence of indeterminacy for some measurements is a characteristic of prevalent interpretations of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle. \stackrel{\rightarrow }{\partial }_{p}-\stackrel{\leftarrow }{\partial }_{p}\stackrel{\rightarrow }{\partial }_{x}) \right)}  \, g  
< 
> .' . \frac{\hbar}{2} \left(m \omega^2 x \stackrel{\rightarrow }{\partial }_{p} - \frac{p}{m} \stackrel{\rightarrow }{\partial }_{x}\right) \cdot W=0 in bra–ket notation:  |\Psi\rangle = \sum_{s_{z1}} \sum_{s_{z2}}\cdots\sum_{s_{zN}}\int_{V_1}\int_{V_2}\cdots\int_{V_N} \mathrm{d}\mathbf{r}_1\mathrm{d}\mathbf{r}_2\cdots\mathrm{d}\mathbf{r}_N \Psi |\mathbf{r}, \mathbf{s_z}\rangle  Sinha, Sukanya; Sorkin, Rafael D. (1991). "A Sum-over-histories Account of an EPR(B) Experiment". Found. Of Phys. Lett. 4 (4): 303–335. Bibcode:1991FoPhL...4..303S. doi:10.1007/BF00665892.  \int_0^T P \;dX = n h   .  
< 
> Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics when the Hamiltonian is quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action. p \approx m v  or Hugh Everett III Manuscript Archive (UC Irvine) – Jeffrey A. Barrett, Peter Byrne, and James O. Weatherall (eds.). 
< 
>  A map from Robert Sobel's novel For Want of a Nail, an artistic illustration of how small events – in this example the branching or point of divergence from our timeline's history is in October 1777 – can profoundly alter the course of history. According to the many-worlds interpretation every event, even microscopic, is a branch point; all possible alternative histories actually exist.[1] \, , Equations[edit]  8 The path integral in quantum-mechanical interpretation ^ Jump up to: a b c d e J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99–124. doi:10.1017/S0305004100000487 
< 
> K(x-y) = \int_0^{\infty} K(x-y,\Tau) W(\Tau) d\Tau For a particle in curved space the kinetic term depends on the position and the above time slicing cannot be applied, this being a manifestation of the notorious operator ordering problem in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation (nonholonomic mapping explained here). Since the role of the observer and measurement per se plays no special role in MWI (measurements are handled as all other interactions are) there is no need for a precise definition of what an observer or a measurement is — just as in Newtonian physics no precise definition of either an observer or a measurement was required or expected. In all circumstances the universal wavefunction is still available to give a complete description of reality.   
< 
> Functional identity[edit] . Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry (for instance, angular or linear momentum). P(E_i) = g(E_i)/(e^{(E-\mu)/kT}+1)\,\!  It is also simple to verify that the matrix f \stackrel{\rightarrow }{\partial }_x g = f \cdot \frac{\partial g}{\partial x}.  U(t) = \mathrm{T}\exp\left({-\frac{i}{\hbar} \int_0^t H(t')\, dt'}\right),  \begin{align} \hat{H} & = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N) \\  | \psi_i\rangle \langle \psi_i| \, 
< 
>  Interpretations of quantum mechanics or  \, , Jump up ^ Wojciech Hubert Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics, 75, pp 715–775, (2003)   
< 
> Debate[edit] which is precisely the partition function of statistical mechanics for the same system at temperature quoted earlier. One aspect of this equivalence was also known to Schrödinger who remarked that the equation named after him looked like the diffusion equation after Wick rotation. 5 Automorphisms \, ,  Other formulations[edit] 
< 
> A visual comparison of the hydrogen spectral series for n1 = 1 to n1 = 6 on a log scale \stackrel{\rightarrow }{\partial }_{p}-\stackrel{\leftarrow }{\partial }_{p}\stackrel{\rightarrow }{\partial }_{x}) \right)}  \, g v t e     \sum_{mn} \psi_m^* A_{mn} \psi_n 
< 
>  i{\partial \over \partial t} \psi_t(x) = \left[-{1\over 2m} {\partial^2 \over \partial x^2} + V(x)\right] \psi_t(x)\, . Branches[edit]  This function must be real, because both P and -iD are Hermitian,  The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves. 
< 
> Statistical structure[edit] Many-worlds in literature and science fiction[edit] which can be identified with i times the k-th classical Fourier component of the Poisson bracket. p and (q or r) = (p and q) or (p and r), 
< 
>  \,  Jump up ^ Penrose, R. The Road to Reality, §21.11 The transformations have the following consequences:  Jump up ^ Pitowsky, I. (2005). "Quantum mechanics as a theory of probability". Eprint arXiv:quant-ph/0510095: 10095. arXiv:quant-ph/0510095. Bibcode:2005quant.ph.10095P. Dirac equation (Original) \gamma^k = \gamma^0 \alpha^k. \, 
< 
>   y' = y Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics when the Hamiltonian is quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action. The contribution of a path is proportional to  e^{i S/\hbar}. while S is the action given by the time integral of the Lagrangian along the path. H \star W = E \cdot W,  
< 
> Further reading[edit] \exp \left (-{b} (x^2+p^2)\right ) = {1\over 1+\hbar^2 ab}   Jump up ^ Harvey R Brown and David Wallace, Solving the measurement problem: de Broglie–Bohm loses out to Everett, Foundations of Physics 35 (2005), pp. 517–540. [3] H_S = H_{0,S} + H_{1, S} ~.     
< 
> Theoretical and experimental justification for the Schrödinger equation Statistical mechanics, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk thermodynamic properties of materials.   \, ,  Background[show] 
< 
> where H is a densely defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and h is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.  [p(t_{1}), p(t_{2})]=i\hbar m\omega\sin(\omega t_{2}-\omega t_{1})  , 8 The measurement process 
< 
> i\hbar\frac{\partial}{\partial t} \Psi = \hat{H}\Psi absorbed dose rate	m2·s-3  For a spatially homogeneous system, where K(x, y) is only a function of (x - y), the integral is a convolution, the final state is the initial state convolved with the propagator.  Property or effect	Nomenclature	Equation {dx \over dt} = {x(t+\epsilon) - x(t) \over \epsilon} Equations[show] The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle: 
< 
> K_{rad}=\frac{\alpha^2 \hbar^2}{2m}\left(\frac{3}{2} - \frac{1}{1+(t/\tau)^2}\right) 14 Further reading Jump up ^ http://cs.bath.ac.uk/ag/p/BVQuantCausEvol.pdf A quantum mechanical particle, on the other hand, can't emit radiation continuously, it can only emit photons. Assuming that the quantum particle started in orbit number n, emitted a photon, then ended up in orbit number m, the energy of the photon is En-Em, which means that its frequency is (En-Em)/h. \begin{bmatrix} [L_z,X+iY] = (X+iY)  
< 
> moment of inertia	kg·m2 \,  The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory and many-body physics.  
< 
> where O_i represents the observer having detected the object system in the ith state. Before the measurement has started the observer states are identical; after the measurement is complete the observer states are orthonormal.[4][7] Thus a measurement defines the branching process: the branching is as well- or ill-defined as the measurement is; the branching is as complete as the measurement is complete – which is to say that the delta function above represents an idealised measurement. Although true "for all practical purposes" in reality the measurement, and hence the branching, is never fully complete, since delta functions are unphysical,[54] When a transformation, like a rotation in physical space, commutes with the Hamiltonian, the transformation is called a symmetry (behind a degeneracy) of the Hamiltonian--the Hamiltonian expressed in terms of rotated coordinates is the same as the original Hamiltonian. This means that the change in the Hamiltonian under the infinitesimal symmetry generator L vanishes,  X_{nm}=X_{mn}^* . Main article: Hydrogen atom q(x)[\phi(y)] = \delta^{(d)}(X-y)Q[\phi(y)] \, where the right hand side is really only the (m - n)'th Fourier component of dA/dJ at the orbit near m to this semiclassical order, not a full well-defined matrix. 
< 
>  D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley,. ISBN 81-7758-293-3. x\rightarrow x+dx = x + {\partial H \over \partial p} dt Time as an operator[edit] Introduction Glossary History 
< 
> Limits of validity[edit] 4 Properties of the theory Jump up ^ D.Deutsch, Int.J.theor.Phys. 24,1 (1985). i\hbar\frac{\partial}{\partial t} \Psi = \hat{H}\Psi  i\hbar{d \over d t}A(t) = [A(t),H_{0}]. 11 References Continuum mechanics, for materials modelled as a continuum, e.g., solids and fluids (i.e., liquids and gases). \psi(y;t+\epsilon) \approx \int \psi(x;t) e^{-{\rm i}\epsilon V(x)} e^{{\rm i}(x-y)^2 \over 2\epsilon} dx  
< 
>    Jump up ^ Simon Saunders, 2004: What is Probability?  
< 
>  ^ Jump up to: a b Peter Byrne, The Many Worlds of Hugh Everett III: Multiple Universes, Mutual Assured Destruction, and the Meltdown of a Nuclear Family, ISBN 978-0-19-955227-6  If an observable is measured and the result is a certain eigenvalue, the corresponding eigenvector is the state of the system immediately after the measurement. The act of measurement in matrix mechanics 'collapses' the state of the system. If one measures two observables simultaneously, the state of the system collapses to a common eigenvector of the two observables. Since most matrices don't have any eigenvectors in common, most observables can never be measured precisely at the same time. This is the uncertainty principle. Splitting the integral into time slices: The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies which are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy. The structure of Q immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a proposition p, the equations 
< 
> {\rm i}{\partial \over \partial t} \psi_t = - {\nabla^2\over 2} \psi_t Commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators. For example, consider the operators x(t1), x(t2), p(t1) and p(t2). The time evolution of those operators depends on the Hamiltonian of the system. Considering the one-dimensional harmonic oscillator, S = | \psi \rangle  \langle \psi  | ,   we get the "master" Schwinger–Dyson equation: G. Birkhoff and J. von Neumann, The Logic of Quantum Mechanics, Annals of Mathematics, Vol. 37, pp. 823–843, 1936. 12 Further reading  i\hbar\frac{d}{dt}A_I(t)=\left[A_I(t),H_0\right].\; 
< 
>  Declaration of completeness of quantum mechanics[edit] \psi(y;t+\epsilon) \approx \int \psi(x;t) e^{-{\rm i}\epsilon V(x)} e^{{\rm i}(x-y)^2 \over 2\epsilon} dx Spin magnitude: A = observables (eigenvalues of operator) Introduction Glossary History 
< 
> In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation. \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ Reception[edit] Harmonic oscillator[edit] For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues ?i, with corresponding eigenvectors ?i. The projection-valued measure associated with A, EA, is then The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities fail.  
< 
> \langle \hat{A} \rangle = \int A(x, p) W(x, p) \, dp \, dx.  The transformations have the following consequences: Jump up ^ Mark A Rubin (2005), There Is No Basis Ambiguity in Everett Quantum Mechanics, Foundations of Physics Letters, Volume 17, Number 4 / August, 2004, pp 323–341  
< 
>  dynamic viscosity	kg·m-1·s-1 So in the relativistic case, the Feynman path-integral representation of the propagator includes paths which go backwards in time, which describe antiparticles. The paths which contribute to the relativistic propagator go forward and backwards in time, and the interpretation of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.  E = {n h \over 2\pi}  ,   Zinn Justin, Jean (2004). Path Integrals in Quantum Mechanics. Oxford University Press. ISBN 0-19-856674-3. A highly readable introduction to the subject. 
< 
>   \Psi = \psi(\mathbf{r}) e^{-iEt/\hbar} 	 \Psi = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N)   \rho = \frac{i\hbar}{2m}(\psi^*\partial_t\psi - \psi\partial_t\psi^*).