Virtual Particles

Prerequisites: Feynman Diagrams. Full path (22 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

also known as Quantum Fluctuations

Intuitive

A quantum vacuum fluctuation is a process where from the vacuum a pair of particle and antiparticle emerge and subsequently vanish again.

The particle and antiparticle pop into existence and then vanish again. The particles that appear in such a process are known as virtual particles.

A quantum vacuum fluctuation does not violate the conservation of any charge since a particle and antiparticle are created at the same time which carry exactly the opposite charges.

The energy that is necessary to create the particle-antiparticle pair is "borrowed" from the vacuum which is in agreement with the "energy-time" uncertainty principle as long as the pair vanishes fast enough again. Similarly, the necessary momentum is borrowed from the vacuum which is allowed thanks to the "momentum-location" uncertainty principle.

Since such processes are allowed in quantum mechanics the vacuum is never really empty. Instead, it is a boiling sea of particles-antiparticle pairs that pop into existence and then vanish again.

These vacuum fluctuation can be observed, for example, in the form of the famous Casimir effect. Here an attractive force between two plates is measured which is a result of the vacuum fluctuations.

Another instance where virtual particles appear is in particle interactions. When two particles, like two electrons collide it can happen that the photon, which is responsible for the interaction, is converted into a electron-positron pair and then back into the photon. These virtual particles must be taken into account in calculations to get the correct results.

Recommended Resources

• Virtual Particles: What are they? | Of Particular Significance
• https://physics.stackexchange.com/questions/34049/what-really-goes-on-in-a-vacuum?lq=1
• http://physics.stackexchange.com/questions/4349/are-w-z-bosons-virtual-or-not/22064#22064
• http://www.mat.univie.ac.at/~neum/physfaq/topics/vacfluc
• Quantum Fluctuations and Their Energy | Of Particular Significance

Concrete

Virtual particles appear in quantum field theory as internal lines in the perturbative expansion of scattering amplitudes. At tree level two particles scatter, for example, by simply exchanging a virtual photon. The first order correction in perturbation theory can be interpreted as an additional process where the photon becomes virtual electron-positron pair and then again back a photon.

The total cross section is the sum over all such possible processes. At higher orders more and more virtual particles can appear. However, for a small coupling like that of QED, each additional order is suppressed by another power of the coupling, and therefore it is a valid approximation to only consider the first few order in perturbation theory.

Recommended Resources

• Learn the Physics of Virtual Particles in Quantum Mechanics
• Top Misconceptions about Virtual Particles
• Explore The Vacuum Fluctuation Myth in Quantum Theory

Talk about quantum fluctuations can be vague. There are really 3 different types of fluctuations: Boltzmann, Vacuum, & Measurement.

• Boltzmann Fluctuations are basically classical: random motions of things lead to unlikely events, even in equilibrium. The *macro*state of a system can be static (equilibrium), but stuff is churning beneath the surface. That allows Boltzmann fluctuations.
• Vacuum Fluctuations are the differences b/w quantum and classical states. Classically-definite observables can have a quantum variance. Zero-point energy, virtual particles, the Lamb shift – all due to vacuum fluctuations. But notice something important. Nothing actually "fluctuates" in vacuum fluctuations! The system can be perfectly static. Just that quantum states are more spread out.
• Measurement Fluctuations are the bridge b/w Boltzmann and Vacuum fluctuations. They occur when you repeatedly observe a quantum system. By itself, a system can be static, but observational outcomes are probabilistic. Observe over and over again, get different results. That makes us confuse (static) vacuum fluctuations with (dynamical) Boltzmann fluctuations. We think what we see is what really is. But quantum mechanics says that what really exists is very different from what we see. In an atom, electrons aren't fluctuating at all.

https://twitter.com/seanmcarroll/status/688483074850422784

The calculational tool represented by Feynman diagrams suggests an often abused picture according to which “real particles interact by exchanging virtual particles”. Many physicists, especially nonexperts, take this picture literally, as something that really and objectively happens in nature. In fact, I have never seen a popular text on particle physics in which this picture was not presented as something that really happens. Therefore, this picture of quantum interactions as processes in which virtual particles exchange is one of the most abused myths, not only in quantum physics, but in physics in general. Indeed, there is a consensus among experts for foundations of QFT that such a picture should not be taken literally. The fundamental principles of quantum theory do not even contain a notion of a “virtual” state. The notion of a “virtual particle” originates only from a specific mathematical method of calculation, called perturbative expansion. In fact, perturbative expansion represented by Feynman diagrams can be introduced even in classical physics [52, 53], but nobody attempts to verbalize these classical Feynman diagrams in terms of classical “virtual” processes. So why such a verbalization is tolerated in quantum physics? The main reason is the fact that the standard interpretation of quantum theory does not offer a clear “canonical” ontological picture of the actual processes in nature, but only provides the probabilities for the final results of measurement outcomes. In the absence of such a “canonical” picture, physicists take the liberty to introduce various auxiliary intuitive pictures that sometimes help them think about otherwise abstract quantum formalism. Such auxiliary pictures, by themselves, are not a sin. However, a potential problem occurs when one forgets why such a picture has been introduced in the first place and starts to think on it too literally. Quantum mechanics: Myths and facts by H. Nikolic

• No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "Virtual Particles"

Abstract

Mathematically, a virtual particle is an internal line of a Feynman diagram, and an internal line is a factor of the propagator. For a scalar field of mass $m$ (signature $(+,-,-,-)$, $\hbar = c = 1$)

$$\langle 0|T\,\phi(x)\phi(y)|0\rangle = D_F(x-y) = \int\frac{d^4p}{(2\pi)^4}\,\frac{i\,e^{-ip\cdot(x-y)}}{p^2 - m^2 + i\epsilon},$$

and the loop momenta are integrated over all of $\mathbb{R}^4$, so $p^2 \neq m^2$ in general. Four-momentum is conserved at every vertex, and the price is that internal lines are off shell. The propagator is a correlation function of the free field. It does not correspond to a state in the Hilbert space, and asymptotic states, the only ones the S-matrix refers to, contain only on-shell particles.

Time-ordered picture. Splitting the propagator in $p^0$,

$$\frac{1}{p^2 - m^2 + i\epsilon} = \frac{1}{2\omega_p}\left(\frac{1}{p^0 - \omega_p + i\epsilon} - \frac{1}{p^0 + \omega_p - i\epsilon}\right), \qquad \omega_p = \sqrt{\vec p^{\,2} + m^2},$$

shows that one covariant diagram is the sum of two time orderings, one with a particle and one with an antiparticle travelling between the vertices. In old-fashioned (time-ordered) perturbation theory the intermediate states are on-shell Fock states, three-momentum is conserved, and energy is not conserved at the vertices; the energy mismatch appears in denominators $1/(E - E_{\text{int}} + i\epsilon)$. Both pictures give the same amplitudes, which is one reason not to read either kind of intermediate line as a literal particle.

Interacting propagator. In the full theory the two-point function has the Källén-Lehmann representation

$$\langle\Omega|T\,\phi(x)\phi(y)|\Omega\rangle = \int_0^\infty d\mu^2\, \rho(\mu^2)\, D_F(x-y;\mu^2), \qquad \rho(\mu^2) = Z\,\delta(\mu^2 - m^2) + \rho_{\text{cont}}(\mu^2),$$

with $\rho \geq 0$ fixed by the states that $\phi$ creates from the vacuum, $0 \leq Z \leq 1$ for a canonically normalized field, $m$ the physical mass, and $\rho_{\text{cont}}$ supported above the multiparticle threshold. An unstable particle does not appear as a delta function; it shows up as a resonance peak in $\rho_{\text{cont}}$, or equivalently as a pole of the analytically continued propagator on the second sheet at $p^2 \approx M^2 - iM\Gamma$. Whether one calls it "real" or "virtual" in a given process then depends on how far $p^2$ is from $M^2$ compared with $M\Gamma$.

Unitarity and cuts. With $S = 1 + iT$, unitarity $S^\dagger S = 1$ gives $-i(T - T^\dagger) = T^\dagger T$, and for forward scattering the optical theorem $2\,\text{Im}\,\mathcal{M}(a\to a) = \sum_f \int d\Pi_f\, |\mathcal{M}(a\to f)|^2$. The Cutkosky rules compute the discontinuity of a loop diagram by replacing the cut propagators with $-2\pi i\,\delta(p^2 - m^2)\,\theta(p^0)$. Internal lines contribute to the imaginary part of an amplitude only in the kinematic regions where they can go on shell, and there they describe real intermediate states.

Virtual particles are therefore terms in the perturbative expansion of correlation functions. Non-perturbative formulations, such as lattice field theory or the axiomatic approach, work directly with correlation functions and the spectrum of the Hamiltonian, and do not contain them.

"[I]n a closed system all quanta can be considered as virtual." Richard Feynman

• Learn the Physics of Virtual Particles in Quantum Mechanics
• Top Misconceptions about Virtual Particles

Why is it interesting?

There are no real one-particle systems in nature, not even few-particle systems. The existence of virtual pairs and of pair fluctuations shows that the days of fixed particle numbers are over." Viki Weisskopf

FAQ

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1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
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13. Special Relativity
14. Lagrangian Formalism
15. Phase Space
16. Classical Field Theory
17. Hamiltonian Formalism
18. Schrödinger Equation
19. Canonical Quantum Mechanics
20. Canonical Quantum Field Theory
21. Perturbation Theory
22. Feynman Diagrams