Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Quantum fluctuation

From Wikipedia, the free encyclopedia
Random change in the energy inside a volume
For related articles, seeQuantum vacuum (disambiguation).

3D visualization of quantum fluctuations of the quantum chromodynamics(QCD) vacuum[1]

Inquantum physics, aquantum fluctuation (also known as avacuum state fluctuation orvacuum fluctuation) is the temporary random change in the amount of energy in a point inspace,[2] as prescribed byWerner Heisenberg'suncertainty principle. They are minute random fluctuations in the values of the fields which represent elementary particles, such aselectric andmagnetic fields which represent theelectromagnetic force carried byphotons,W and Z fields which carry theweak force, andgluon fields which carry thestrong force.[3]

Theuncertainty principle states the uncertainty inenergy andtime can be related by[4]ΔEΔt≥12ℏ {\displaystyle \Delta E\,\Delta t\geq {\tfrac {1}{2}}\hbar ~}, where⁠1/2⁠ħ ≈5.27286×10−35 J⋅s. This means that pairs of virtual particles with energyΔE{\displaystyle \Delta E} and lifetime shorter thanΔt{\displaystyle \Delta t} are continually created and annihilated inempty space. Although the particles are not directly detectable, the cumulative effects of these particles are measurable. For example, without quantum fluctuations, the"bare" mass and charge of elementary particles would be infinite; fromrenormalization theory the shielding effect of the cloud of virtual particles is responsible for the finite mass and charge of elementary particles.

Another consequence is theCasimir effect. One of the first observations which was evidence forvacuum fluctuations was theLamb shift in hydrogen. In July 2020, scientists reported that quantum vacuum fluctuations can influence the motion of macroscopic, human-scale objects by measuring correlations below thestandard quantum limit between the position/momentum uncertainty of the mirrors ofLIGO and the photon number/phase uncertainty of light that they reflect.[5][6][7]

Field fluctuations

[edit]

Inquantum field theory, fields undergo quantum fluctuations. A reasonably clear distinction can be made between quantum fluctuations andthermal fluctuations of aquantum field (at least for a free field; for interacting fields,renormalization substantially complicates matters). An illustration of this distinction can be seen by considering relativistic and non-relativistic Klein–Gordon fields:[8] For therelativistic Klein–Gordon field in thevacuum state, we can calculate the propagator that we would observe a configurationφt(x){\displaystyle \varphi _{t}(x)} at a timet in terms of itsFourier transformφ~t(k){\displaystyle {\tilde {\varphi }}_{t}(k)} to be

ρ0[φt]=exp⁡[−itℏ∫d3k(2π)3φ~t∗(k)|k|2+m2φ~t(k)].{\displaystyle \rho _{0}[\varphi _{t}]=\exp {\left[-{\frac {it}{\hbar }}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}(k){\sqrt {|k|^{2}+m^{2}}}\,{\tilde {\varphi }}_{t}(k)\right]}.}

In contrast, for thenon-relativistic Klein–Gordon field at non-zero temperature, theGibbs probability density that we would observe a configurationφt(x){\displaystyle \varphi _{t}(x)} at a timet{\displaystyle t} is

ρE[φt]=exp⁡[−H[φt]/kBT]=exp⁡[−1kBT∫d3k(2π)3φ~t∗(k)12(|k|2+m2)φ~t(k)].{\displaystyle \rho _{E}[\varphi _{t}]=\exp {\big [}-H[\varphi _{t}]/k_{\text{B}}T{\big ]}=\exp {\left[-{\frac {1}{k_{\text{B}}T}}\int {\frac {d^{3}k}{(2\pi )^{3}}}{\tilde {\varphi }}_{t}^{*}(k){\frac {1}{2}}\left(|k|^{2}+m^{2}\right)\,{\tilde {\varphi }}_{t}(k)\right]}.}

These probability distributions illustrate that every possible configuration of the field is possible, with the amplitude of quantum fluctuations controlled by thePlanck constantℏ{\displaystyle \hbar }, just as the amplitude of thermal fluctuations is controlled bykBT{\displaystyle k_{\text{B}}T}, wherekB is theBoltzmann constant. Note that the following three points are closely related:

  1. the Planck constant has units ofaction (joule-seconds) instead of units of energy (joules),
  2. the quantum kernel is|k|2+m2{\displaystyle {\sqrt {|k|^{2}+m^{2}}}} instead of12(|k|2+m2){\displaystyle {\tfrac {1}{2}}{\big (}|k|^{2}+m^{2}{\big )}} (the relativistic quantum kernel is nonlocal differently from the non-relativistic classicalheat kernel, but it is causal),[citation needed]
  3. the quantum vacuum state isLorentz-invariant (although not manifestly in the above), whereas the classical thermal state is not (both the non-relativistic dynamics and the Gibbs probability density initial condition are not Lorentz-invariant).

Aclassical continuous random field can be constructed that has the same probability density as the quantum vacuum state, so that the principal difference from quantum field theory is the measurement theory (measurement in quantum theory is different from measurement for a classical continuous random field, in that classical measurements are always mutually compatible – in quantum-mechanical terms they always commute).

Quantum fluctuations as loop effects

[edit]
Tree-level electron propagator

In the language ofFeynman diagrams, quantum fluctuations enter at the level of loop diagrams. Inquantum electrodynamics, for example, the electron self energy diagram (to the right, below) would constitute quantum fluctuations in relation to the electron propagator (to the right, above).

Loop correction to electron propagator; referred to as "electron self-energy"

These loop diagrams are initially problematic; they introduce an integral over the loop momentum (in this casek{\displaystyle k}) from−∞{\displaystyle -\infty } to∞{\displaystyle \infty }, allowing contributions from arbitrarily large momenta. In the case of the electron self energy, the integral is logarithmically divergent and leads to an infiniteamplitude. This problem is addressed byrenormalizing the theory, which corresponds to absorbing the infinity into the mass parameter in the case of the electron self energy. In this example, we write the amplitude of the self energy diagram asPe(p)(−iΣ2(p))Pe(p){\displaystyle \textstyle P_{e}(p)(-i\Sigma _{2}({\cancel {p}}))P_{e}(p)}, wherePe(p)=ip−m0{\displaystyle \textstyle P_{e}(p)={\frac {i}{{\cancel {p}}-m_{0}}}} is the electron propagator and−iΣ2(p){\displaystyle -i\Sigma _{2}({\cancel {p}})} represents the loop component. By generalizing the loop to aone particle irreducible (1PI) diagram−iΣ(p){\displaystyle -i\Sigma ({\cancel {p}})}, we can write the full propagator as a sum of 1PI diagrams:

ip−m0+ip−m0(−iΣ)ip−m0+ip−m0(−iΣ)ip−m0(−iΣ)ip−m0+⋯=ip−m0+ip−m0(Σp−m0)+ip−m0(Σp−m0)2+⋯{\textstyle {\begin{aligned}&\textstyle {\frac {i}{{\cancel {p}}-m_{0}}}+{\frac {i}{{\cancel {p}}-m_{0}}}(-i\Sigma ){\frac {i}{{\cancel {p}}-m_{0}}}+{\frac {i}{{\cancel {p}}-m_{0}}}(-i\Sigma ){\frac {i}{{\cancel {p}}-m_{0}}}(-i\Sigma ){\frac {i}{{\cancel {p}}-m_{0}}}+\cdots \\=&\textstyle {\frac {i}{{\cancel {p}}-m_{0}}}+{\frac {i}{{\cancel {p}}-m_{0}}}\left({\frac {\Sigma }{{\cancel {p}}-m_{0}}}\right)+{\frac {i}{{\cancel {p}}-m_{0}}}\left({\frac {\Sigma }{{\cancel {p}}-m_{0}}}\right)^{2}+\cdots \end{aligned}}}

This is just a geometric series,∑arn{\displaystyle \textstyle \sum ar^{n}}; the solution isa/(1−r){\displaystyle a/(1-r)}, orip−m0−Σ{\displaystyle {\frac {i}{{\cancel {p}}-m_{0}-\Sigma }}}This is the step in which the infinity (Σ{\displaystyle \Sigma }) is absorbed into the mass parameter:m0{\displaystyle m_{0}} is in fact not the observable mass, but simply the mass parameter in theQEDLagrangian; the observable (or "physical") mass is defined as the pole mass (the mass at which the propagator has apole), which in this case ism≡m0+Σ{\displaystyle m\equiv m_{0}+\Sigma }. We know thatΣ{\displaystyle \Sigma } is infinite (recall, we said it was logarithmically divergent), andm0{\displaystyle m_{0}} is unobservable -- this allows us to conclude thatm0{\displaystyle m_{0}} is itself must be infinite so that the summ0+Σ{\displaystyle m_{0}+\Sigma } is regular.

Quantum fluctuations and effective field theories

[edit]

The goal of aneffective field theory is to describe the effects of high-energy physics at low energies. Quantum (field) fluctuations play a crucial role in formulating theeffective actionSeff=∫dDx Leff{\displaystyle S_{\text{eff}}=\int d^{D}x\ {\mathcal {L}}_{\text{eff}}}, which addresses this goal exactly. Specifically, the frequently-used derivative expansion[9] involves splitting a quantum fieldϕ(x){\displaystyle \phi (x)} into a classical background fieldϕcl(x){\displaystyle \phi _{\text{cl}}(x)} and a quantum field encompassing high-energy fluctuations,ω(x){\displaystyle \omega (x)}, as inϕ(x)=ϕcl(x)+ω(x){\displaystyle \phi (x)=\phi _{\text{cl}}(x)+\omega (x)}.

A central idea in the study of effective field theories involves the fact that thegenerating functionalZ[J]{\displaystyle Z[J]} -- an abstract quantity which produces correlation functions via the relationship⟨ϕ(x1)⋯ϕ(xn)⟩=1Z[0](−iδδJ(x1))⋯(−iδδJ(xn))Z[J]|J=0{\textstyle \langle \phi (x_{1})\cdots \phi (x_{n})\rangle ={\frac {1}{Z[0]}}\left(-i{\frac {\delta }{\delta J(x_{1})}}\right)\cdots \left(-i{\frac {\delta }{\delta J(x_{n})}}\right)Z[J]{\bigg |}_{J=0}} -- includes an integral over field configurations,Z[J]=∫Dϕ exp⁡(iS+i∫dDx ϕ(x)J(x)){\displaystyle Z[J]=\int {\mathcal {D}}\phi \ \exp \left(iS+i\int d^{D}x\ \phi (x)J(x)\right)}. If our goal is to describe high-energy physics at low energies, we can splitϕ(x)=ϕcl(x)+ω(x){\displaystyle \phi (x)=\phi _{\text{cl}}(x)+\omega (x)} as prescribed before and simply integrate out theω(x){\displaystyle \omega (x)} fields. The result of this integration allows us to obtain the effective Lagrangian,Leff=L0+(sum of connected Feynman diagrams){\textstyle \textstyle {\mathcal {L}}_{\text{eff}}={\mathcal {L}}_{0}+({\text{sum of connected Feynman diagrams}})}, withL0{\displaystyle {\mathcal {L}}_{0}} being the expression for the original Lagrangian. The term(sum of connected Feynman diagrams){\displaystyle \textstyle ({\text{sum of connected Feynman diagrams}})} precisely accounts for the effects of high-energy fluctuations at low energies.

See also

[edit]

References

[edit]
  1. ^"Derek Leinweber".www.physics.adelaide.edu.au. Retrieved13 December 2020.
  2. ^Pahlavani, Mohammad Reza (2015).Selected Topics in Applications of Quantum Mechanics. BoD. p. 118.ISBN 9789535121268.
  3. ^Pagels, Heinz R. (2012).The Cosmic Code: Quantum Physics as the Language of Nature. Courier Corp. pp. 274–278.ISBN 9780486287324.
  4. ^Mandelshtam, Leonid;Tamm, Igor (1945)."Соотношение неопределённости энергия-время в нерелятивистской квантовой механике" [The uncertainty relation between energy and time in non-relativistic quantum mechanics].Izv. Akad. Nauk SSSR (Ser. Fiz.) (in Russian).9:122–128. English translation:"The uncertainty relation between energy and time in non-relativistic quantum mechanics".J. Phys. (USSR).9:249–254. 1945.
  5. ^"Quantum fluctuations can jiggle objects on the human scale".phys.org. Retrieved15 August 2020.
  6. ^"LIGO reveals quantum correlations at work in mirrors weighing tens of kilograms".Physics World. 1 July 2020. Retrieved15 August 2020.
  7. ^Yu, Haocun; McCuller, L.; Tse, M.; Kijbunchoo, N.; Barsotti, L.; Mavalvala, N. (July 2020)."Quantum correlations between light and the kilogram-mass mirrors of LIGO".Nature.583 (7814):43–47.arXiv:2002.01519.Bibcode:2020Natur.583...43Y.doi:10.1038/s41586-020-2420-8.ISSN 1476-4687.PMID 32612226.S2CID 211031944.
  8. ^Morgan, Peter (2001). "A classical perspective on nonlocality in quantum field theory".arXiv:quant-ph/0106141.
  9. ^Massó, Eduard; Rota, Francesc (14 January 2002)."Summing the derivative expansion of the effective action".Nuclear Physics B.620 (3):566–578.doi:10.1016/S0550-3213(01)00537-5.ISSN 0550-3213.
Theories
Models
Regular
Low dimensional
Conformal
Supersymmetric
Superconformal
Supergravity
Topological
Particle theory
Related
Background
Fundamentals
Formulations
Equations
Interpretations
Experiments
Science
Technology
Extensions
Related
Retrieved from "https://en.wikipedia.org/w/index.php?title=Quantum_fluctuation&oldid=1337340659"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2026 Movatter.jp