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Asher Anderson
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Foundations Of Quantum Gravity

More than two decades of research have resulted in remarkable theoretical progress and experimental capabilities that now enable us to revisit the very foundations of quantum theory. To make a cartographic analogy, our present understanding of quantum mechanics is like an island containing still uncharted regions and with indistinct coastlines; even less is known of what may lie beyond the surrounding seas. This Nature Physics Insight covers some of the exploratory attempts to improve our map of the quantum world.

Foundations of Quantum Gravity

In physics, as in life, it's always good to look at things from different perspectives. googletag.cmd.push(function() googletag.display('div-gpt-ad-1449240174198-2'); ); Since the beginning of quantum physics, how light moves and interacts with matter around it has mostly been described and understood mathematically through the lens of its energy. In 1900, Max Planck used energy to explain how light is emitted by heated objects, a seminal study in the foundation of quantum mechanics. In 1905, Albert Einstein used energy when he introduced the concept of photon.

An international team of physicists led by Michaël Lobet, a research associate at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) and Eric Mazur, the Balkanski Professor of Physics and Applied Physics at SEAS, are re-examining the foundations of quantum physics from the perspective of momentum and exploring what happens when the momentum of light is reduced to zero.

But a century after Planck and Einstein, a new class of metamaterials is raising questions regarding these fundamental phenomena. These metamaterials have a refractive index close to zero, meaning that when light travels through them, it doesn't travel like a wave in phases of crests and troughs. Instead, the wave is stretched out to infinity, creating a constant phase. When that happens, many of the typical processes of quantum mechanics disappear, including atomic recoil.

"As it can be seen, this work interrogates fundamental laws of quantum mechanics and probes the limits of wave-corpuscle duality," said co-author Iñigo Liberal, of the Public University of Navarre in Pamplona, Spain.

"These new theoretical results shed new light on near-zero refractive index photonics from a momentum perspective," said Mazur. "It provides insights in the understanding of light-matter interactions in systems with a low- refraction index, which can be useful for lasing and quantum optics applications."

The research could also shed light on other applications, including quantum computing, light sources that emit a single photon at a time, the lossless propagation of light through a waveguide and more.

The team next aims to revisit other foundational quantum experiments in these materials from a momentum perspective. After all, even though Einstein didn't predict near-zero refractive index materials, he did stress the importance of momentum. In his seminal 1916 paper on fundamental radiative processes, Einstein insisted that from a theoretical point of view, energy and momentum "should be considered on a completely equal footing since energy and momentum are linked in the closest possible way."

In classical physics, with any physical system is associated a statespace, which represents the totality of possible ways of assigningvalues to the dynamical variables that characterize the state of thesystem. For systems of a great many degrees of freedom, a completespecification of the state of the system may be unavailable orunwieldy; classical statistical mechanics deals with such a situationby invoking a probability distribution over the state space of thesystem. A probability distribution that assigns any probability otherthan one or zero to some physical quantities is regarded as anincomplete specification of the state of the system. In quantummechanics, things are different. There are no quantum states thatassign definite values to all physical quantities, and probabilitiesare built into the standard formulation of the theory.

Construction of a quantum theory of a physical system proceeds byfirst associating the dynamical degrees of freedom withoperators. These are mathematical objects on which operationsof multiplication and addition are defined, as well as multiplicationby real and complex numbers. Another way of saying this is that theset of operators forms an algebra. Typically, it is said thatan operator represents an observable, and the result of anexperiment on a system is said to yield a value for some observable.Two or more observables are said to be compatible if there issome possible experiment that simultaneously yields values for all ofthem. Others require mutually exclusive experiments; these are said tobe incompatible.

The recipe for constructing a quantum theory of a given physicalsystems prescribes algebraic relations between the operatorsrepresenting the dynamical variables of the system. Compatibleobservables are associated with operators that commute with eachother. Operators representing conjugate variables are required tosatisfy what are called the canonical commutation relations.If \(q\) is some coordinate, and \(p\) its conjugatemomentum, the operators \(Q\) and \(P\) representing themare required to not commute. Instead, the difference between\(PQ\) and \(QP\) is required to be a multiple of theidentity operator (that is, the operator \(I\) that satisfies,for all operators \(A\), \(IA = AI).\)

A quantum state is a specification, for every experiment thatcan be performed on the system, of probabilities for thepossible outcomes of that experiment. These can be summed up as anassignment of an expectation value to each observable. These statesare required to be linear. This means that, if an operator\(C\), corresponding to some observable, is the sum of operators\(A\) and \(B\), corresponding to other observables, thenthe expectation value that a quantum state assigns to \(C\) mustbe the sum of the expectation values assigned to \(A\) and\(B\). This is a nontrivial constraint, as it is required to holdwhether or not the observables represented are compatible. A quantumstate, therefore, relates expectation values for quantities yielded byincompatible experiments.

Incompatible observables, represented by noncommuting operators, giverise to uncertainty relations; see the entry on the uncertainty principle. These relations entail that there are no quantum states that assigndefinite values to the observables that satisfy them, and place boundson how close they can come to be simultaneously well-defined in anyquantum state.

For any two distinct quantum states, \(\rho\), \(\omega\), and anyreal number between 0 and 1, there is a corresponding mixedstate. The probability assigned to any experimental outcome bythis mixed state is \(p\) times the probability it is assigned by\(\rho\) plus \(1-p\) times the probability assigned to it by\(\omega\). One way to physically realize the preparation of a mixedstate is to employ a randomizing device, for example, a coin withprobability \(p\) of landing heads and probability \(1-p\) oflanding tails, and to use it to choose between preparing state\(\rho\) and preparing state \(\omega\). We will see another way toprepare a mixed state after we have discussed entanglement, in section3. A state that is not a mixture of any two distinct states is calleda pure state.

It is both useful and customary, though not strictly necessary, toemploy a Hilbert space representation of a quantum theory. Insuch a representation, the operators corresponding to observables arerepresented as acting on elements of an appropriately constructedHilbert space (see the entry on quantum mechanics for details). Usually, the Hilbert space representation isconstructed in such a way that vectors in the space represent purestates; such a representation is called an irreduciblerepresentation. Irreducible representations, in which mixedstates are also represented by vectors, are also possible.

A Hilbert space is a vector space. This means that, for any twovectors \(\psi\rangle\), \(\phi\rangle\) , in the space,representing pure states, and any complex numbers \(a\),\(b\), there is another vector, \(a \psi\rangle + b\phi\rangle\), that also represents a pure state. This is called asuperposition of the states represented by \(\psi\rangle\)and \(\phi\rangle\) . Any vector in a Hilbert space can be written asa superposition of other vectors in infinitely many ways. Sometimes,in discussing the foundations of quantum mechanics, authors fall intotalking as if some state are superpositions and others are not. Thisis simply an error. Usually what is meant is that some states yielddefinite values for macroscopic observables, and others cannot bewritten in any way that is not a superposition of macroscopicallydistinct states.

The noncontroversial operational core of quantum theory consists ofrules for identifying, for any given system, appropriate operatorsrepresenting its dynamical quantities. In addition, there areprescriptions for evolving the state of system when it is acted uponby specified external fields or subjected to various manipulations(see section 1.3). Application of quantum theory typically involves a distinctionbetween the system under study, which is treated quantum mechanically,and experimental apparatus, which is not. This division is sometimesknown as the Heisenberg cut.

Whether or not we can expect to be able to go beyond thenoncontroversial operational core of quantum theory, and take it to bemore than a means for calculating probabilities of outcomes ofexperiments, remains a topic of contemporary philosophicaldiscussion.

Quantum mechanics is usually taken to refer to the quantizedversion of a theory of classical mechanics, involving systems with afixed, finite number of degrees of freedom. Classically, a field, suchas, for example, an electromagnetic field, is a system endowed withinfinitely many degrees of freedom. Quantization of a field theorygives rise to a quantum field theory. The chief philosophicalissues raised by quantum mechanics remain when the transition is madeto a quantum field theory; in addition, new interpretational issuesarise. There are interesting differences, both technical andinterpretational, between quantum mechanical theories and quantumfield theories; for an overview, see the entries on quantum field theory and quantum theory: von Neumann vs. Dirac. 041b061a72


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