nLab vacuum stability

Idea

In quantum field theory, a given vacuum state is called stable if in a suitable sense it does not evolve into or from any orthogonal state.

In perturbative quantum field theory with specified S-matrix $\mathcal{S}(g S_{int})$ it makes sense to say that a vacuum state is stable if there is vanishing quantum probability for the vacuum state to scatter into a non-vacuum state, or for a non-vacuum state to scatter into the vacuum state. (e.g. Nikolic 01, p. 6)

Definition

Definition

(vacuum stability)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to (this def.), let $\mathcal{S}$ be a corresponding S-matrix scheme according to (this. def.), and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle$ be a local observable, regarded as an adiabatically switched interaction action functional.

We say that the given Hadamard vacuum state (this prop.)

$$\langle - \rangle \;\colon\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g, j ] ] \longrightarrow \mathbb{C}[ [ \hbar, g, j ] ]$$

is stable with respect to the interaction $S_{int}$, if for all elements of the Wick algebra

$$A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g] ]$$

we have

$$\left\langle A \mathcal{S}(g S_{int}) \right\rangle \;=\; \left\langle \mathcal{S}(g S_{int}) \right\rangle \, \left\langle A \right\rangle \phantom{AA} \text{and} \phantom{AA} \left\langle \mathcal{S}(g S_{int})^{-1} A \right\rangle \;=\; \frac{1} { \left\langle \mathcal{S}(g S_{int}) \right\rangle } \left\langle A \right\rangle$$

Properties

Example

(scattering amplitudes as vacuum expectation values of interacting field observables)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a relativistic free vacuum according to this def., let $\mathcal{S}$ be a corresponding S-matrix scheme according to this def., and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a local observable regarded as an adiabatically switched interaction-functional, such that the vacuum state is stable with respect to $g S_{int}$ (def. ).

Consider local observables

$$\array{ A_{in,1}, \cdots, A_{in , n_{in}}, \\ A_{out,1}, \cdots, A_{out, n_{out}} } \;\;\in\;\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]$$

whose spacetime support satisfies the following causal ordering:

$$A_{out, i_{out} } {\gt\!\!\!\!\lt} A_{out, j_{out}} \phantom{AAA} A_{out, i_{out} } {\vee\!\!\!\wedge} S_{int} {\vee\!\!\!\wedge} A_{in, i_{in}} \phantom{AAA} A_{in, i_{in} } {\gt\!\!\!\!\lt} A_{in, j_{in}}$$

for all $1 \leq i_{out} \lt j_{out} \leq n_{out}$ and $1 \leq i_{in} \lt j_{in} \leq n_{in}$.

Then the vacuum expectation value of the Wick algebra-product of the corresponding interacting field observables (this def.) is

$$\begin{aligned} & \left\langle {\, \atop \,} (A_{out, 1})_{int} \cdots (A_{out,n_{out}})_{int} \, (A_{in, 1})_{int} \cdots (A_{in,n_{in}})_{int} {\, \atop \,} \right\rangle \\ & = \left\langle {\, \atop \,} A_{out,1} \cdots A_{out,n_{out}} \right| \; \mathcal{S}(g S_{int}) \; \left| A_{in,1} \cdots A_{in, n_{in}} {\, \atop \,} \right\rangle \\ & \coloneqq \frac{1}{ \left\langle \mathcal{S}(g S_{int}) \right\rangle } \left\langle {\, \atop \,} A_{out,1} \cdots A_{out,n_{out}} \; \mathcal{S}(g S_{int}) \; A_{in,1} \cdots A_{in, n_{in}} {\, \atop \,} \right\rangle \,. \end{aligned}$$

These vacuum expectation values are interpreted, in the adiabatic limit where $g_{sw} \to 1$, as scattering amplitudes (see this remark).

For proof see at S-matrix, this prop..

Examples

• Higgs field – Mass and Vacuum (in-)stability

Related concepts

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

• Günter Scharf, Vacuum stability in quantum field theory, Nuovo Cim. A109 (1996) 1605-1607 (spire:432208)

Discussion for QED:

• Hrvoje Nikolić, Physical stability of the QED vacuum, 2001 (arXiv:hep-ph/0105176)

Discussion for Higgs field:

• J.R. Espinosa, G. Giudice, A. Riotto, Cosmological implications of the Higgs mass measurement, JCAP 0805:002, 2008 (arXiv:0710.2484)

• John Ellis, J.R. Espinosa, G.F. Giudice, A. Hoecker, A. Riotto, The Probable Fate of the Standard Model, Phys. Lett. B679:369-375, 2009 (arXiv:0906.0954)

• Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, Gian Giudice, Filippo Sala, Alberto Salvio, Alessandro Strumia, Investigating the near-criticality of the Higgs boson (arXiv:1307.3536)

• Anson Hook, John Kearney, Bibhushan Shakya, Kathryn M. Zurek, Probable or Improbable Universe? Correlating Electroweak Vacuum Instability with the Scale of Inflation, J. High Energ. Phys. (2015) 2015: 61 (arXiv:1404.5953)

• Jose R. Espinosa, Gian F. Giudice, Enrico Morgante, Antonio Riotto, Leonardo Senatore, Alessandro Strumia, Nikolaos Tetradis, The cosmological Higgstory of the vacuum instability (arXiv:1505.04825)

• William E. East, John Kearney, Bibhushan Shakya, Hojin Yoo, Kathryn M. Zurek, Spacetime Dynamics of a Higgs Vacuum Instability During Inflation, Phys. Rev. D 95, 023526 (2017) (arXiv:1607.00381)

• Gordon Kane, Exciting Implications of LHC Higgs Boson Data (arXiv:1802.05199)