Galilean Invariance and the General Covariance of Nonrelativistic Laws

Rosen, Gerald

doi:10.1119/1.1986618

Cited by 54 publications

(45 citation statements)

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“…The number of such factors in the interaction vertices in Eqs. (13), (14), (15), and (16) would be 2, 1, 2, and 2, respectively. These factors can be absorbed into the coupling constants.…”

Section: Feynman Rulesmentioning

confidence: 99%

“…Galilean invariance is a possible space-time symmetry of a nonrelativistic theory that requires exact mass conservation [15]. The mass that must be conserved is the kinetic mass, which is the mass that appears in the denominator of the kinetic energy.…”

Section: A Galilean Invariancementioning

confidence: 99%

“…The interaction vertices in Eqs. (13), (14), (15), and (16) would be multiplied by Λ raised to the powers 3 − d, (3 − d)/2, 3 − d, and 3 − d, respectively. In a Green function, the net effect of these powers of Λ is a factor of Λ 3−d for every loop integral and an overall multiplicative factor of Λ 3−d raised to some power.…”

Section: Feynman Rulesmentioning

confidence: 99%

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Galilean-invariant effective field theory for theX(3872)

Braaten

2015

Phys. Rev. D

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XEFT is a low-energy effective field theory for charm mesons and pions that provides a systematically improvable description of the X(3872) resonance. A Galilean-invariant formulation of XEFT is introduced to exploit the fact that mass is very nearly conserved in the transition D * 0 → D 0 π 0 . The transitions D * 0 → D 0 π 0 and X → D 0D0 π 0 are described explicitly in XEFT. The effects of the decay D * 0 → D 0 γ and of short-distance decay modes of the X(3872), such as J/ψ π + π − , can be taken into account by using complex on-shell renormalization schemes for the D * 0 propagator and for the D * 0D0 propagator in which the positions of their complex poles are specified. Galileaninvariant XEFT is used to calculate the D * 0D0 scattering length to next-to-leading order. Galilean invariance ensures the cancellation of ultraviolet divergences without the need for truncating an expansion in powers of the ratio of the pion and charm meson masses.

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“…The number of such factors in the interaction vertices in Eqs. (13), (14), (15), and (16) would be 2, 1, 2, and 2, respectively. These factors can be absorbed into the coupling constants.…”

Section: Feynman Rulesmentioning

confidence: 99%

Section: A Galilean Invariancementioning

confidence: 99%

Section: Feynman Rulesmentioning

confidence: 99%

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Galilean-invariant effective field theory for theX(3872)

Braaten

2015

Phys. Rev. D

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“…The discrete version of the Schrödinger equation with diagonal (on-site) disorder is also the prototypal model of Anderson localization [31], whereas the discrete Schrödinger equation with a sinusoidal potential describes a crystal electron in a uniform magnetic (Harper model) which yields a fractal structure of energy spectrum (Hofstadter butterfly [32]). Another important effect of space discretization, which has not received so far much attention, is breakdown of the covariance of the Schrödinger equation for Galilean boosts [33]. Such a result has a deep physical consequence and indicates that discrete wave dynamics is distinct in different inertial reference frames.…”

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confidence: 99%

Bound states of moving potential wells in discrete wave mechanics

Longhi¹

2017

EPL

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-Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schrödinger equation. While for a vanishing lattice spacing wave evolution of the continuous Schrödinger equation is retrieved, spatial discretization and lattice effects can deeply modify wave dynamics. Here we discuss implications of breakdown of exact Galilean invariance of the discrete Schrödinger equation on the bound states sustained by a smooth potential well which is uniformly moving on the lattice with a drift velocity v. While in the continuous limit the number of bound states does not depend on the drift velocity v, as one expects from the covariance of ordinary Schrödinger equation for a Galilean boost, lattice effects can lead to a larger number of bound states for the moving potential well as compared to the potential well at rest. Moreover, for a moving potential bound states on a lattice become rather generally quasi-bound (resonance) states.Introduction. -One of the cornerstones of nonrelativistic quantum mechanics is the Schrödinger equation, which describes the temporal evolution of a particle wave function based on its initial state. Traditionally, in wave mechanics space and time are considered continuous. However, on several occasions authors have debated about the nature of space-time manifold and the possibility of considering variant forms of the Schrödinger equation in which the wave function is defined on discrete lattice sites of space, time, or space-time, instead of on the space-time continuum [1][2][3][4][5][6][7][8][9][10][11][12]. Fundamental limits to a minimum measurable length were suggested in the early days of quantum physics, notably by Heisenberg, and appear, for example, in modern theories of loop quantum gravity, where spacetime looks granular [13]. A phenomenological approach to account for a minimum length scale is to consider an extension of the uncertainty principle by deforming the canonical commutation relations of position and momentum operators [14], leading to an extended form of the Schödinger equation [15,16]. Other simple models consider the non-relativistic Schrödinger equation defined on a discrete lattice [7][8][9][10][11][12][17][18][19][20][21][22], leading to so-called discrete wave mechanics [7] or discrete quantum mechanics [20]. While earlier models of discrete wave mechanics [7][8][9][10][11][12] did not find great relevance as foundational theories, in

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“…Second, we can accept Geometry 1, but then accept that the sensitivity of my wrist-watch to accelerations had longsince undermined any possibility of interpreting GR as a geometrical theory. 13 I feel confident that most readers would choose the former.…”

Section: The Clock Hypothesismentioning

confidence: 99%

Flavour-Oscillation Clocks and the Geometricity of General Relativity

Knox

2010

The British Journal for the Philosophy of Science

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I look at the 'flavour-oscillation clocks' proposed by D.V. Ahluwalia, and two arguments of his suggesting that such clocks might behave in a way that threatens the geometricity of general relativity (GR). The first argument states that the behaviour of these clocks in the vicinity of a rotating gravitational source implies a non-geometric element of gravity. I argue that the phenomenon is best seen as an instance of violation of the 'clock hypothesis', and therefore does not threaten the geometrical nature of gravitation. Ahluwalia's second argument, for the 'incompleteness' of general relativity, involves the idea that flavour-oscillation clocks can detect constant gravitational potentials. I argue that the purported 'incompleteness-establishing' result is in fact one that applies to all clocks. It is entirely derivable from GR, does not result in the observability of the potential, and is not at odds with any of GR's foundations.

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Galilean Invariance and the General Covariance of Nonrelativistic Laws

Cited by 54 publications

References 0 publications

Galilean-invariant effective field theory for theX(3872)

Galilean-invariant effective field theory for theX(3872)

Bound states of moving potential wells in discrete wave mechanics

Flavour-Oscillation Clocks and the Geometricity of General Relativity

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