Theta vacuum
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In quantum field theory, the theta vacuum is the semi-classical
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
of non- abelian Yang–Mills theories specified by the vacuum angle ''θ'' that arises when the state is written as a superposition of an infinite set of topologically distinct vacuum states. The dynamical effects of the vacuum are captured in the Lagrangian formalism through the presence of a ''θ''-term which in quantum chromodynamics leads to the fine tuning problem known as the strong CP problem. It was discovered in 1976 by Curtis Callan, Roger Dashen, and
David Gross David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. ...
, and independently by
Roman Jackiw Roman Wladimir Jackiw (; born 8 November 1939) is a theoretical physicist and Dirac Medallist. Born in Lubliniec, Poland in 1939 to a Ukrainian family, the family later moved to Austria and Germany before settling in New York City when Jackiw w ...
and Claudio Rebbi.


Yang–Mills vacuum


Topological vacua

The semi-classical vacuum structure of non-abelian Yang–Mills theories is often investigated in Euclidean spacetime in some fixed gauge such as the
temporal gauge In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
A_0 = 0. Classical ground states of this theory have a vanishing field strength tensor which corresponds to pure gauge configurations A_i = i\Omega \nabla_i \Omega^, where at each point in spacetime \Omega(x) is some gauge transformation belonging to the non-abelian gauge
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
G. To ensure that the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
is finite, \Omega(x) approaches some fixed value \Omega_\infty as , \boldsymbol x, \rightarrow \infty. Since all points at spatial infinity now behave as a single new point, the spatial manifold \mathbb R^3 behaves as a 3-sphere S^3 = \mathbb R^3 \cup \ so that every pure gauge choice for the gauge field is described by a mapping \Omega(x): S^3 \rightarrow G. When every ground state configuration can be smoothly transformed into every other ground state configuration through a smooth gauge transformation then the theory has a single vacuum state, but if there are topologically distinct configurations then it has multiple vacua. This is because if there are two different configurations that are not smoothly connected, then to transform one into the other one must pass through a configuration with non-vanishing field strength tensor, which will have non-zero energy. This means that there is an energy barrier between the two vacua, making them distinct. The question of whether two gauge configurations can be smoothly deformed into each other is formally described by the homotopy group of the mapping \Omega(x): S^3 \rightarrow G. For example, the gauge group G=\text(2) has an underlying manifold of S^3 so that the mapping is \Omega(x):S^3 \rightarrow S^3, which has a homotopy group of \pi_3(\text(2)) = \mathbb Z. This means that every mapping has some integer associated with it called its
winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
, also known as its
Pontryagin index In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundl ...
, with it roughly describing to how many times the spatial S^3 is mapped onto the group S^3, with negative windings occurring due to a flipped orientation. Only mappings with the same winding number can be smoothly deformed into each other and are said to belong to the same homotopy class. Gauge transformations which preserve the winding number are called small gauge transformations while ones that change the winding number are called large gauge transformations. For other non-abelian gauge groups G it is sufficient to focus on one of their \text(2) subgroups, ensuring that \pi_3(G) = \mathbb Z. This is because every mapping of S^3 onto G can be continuously deformed into a mapping onto an \text(2) subgroup of G, a result that follows from Botts theorem. This is in contrast to abelian gauge groups where every mapping S^3\rightarrow \text(1) can be deformed to the constant map and so there is a single connected vacuum state. For a gauge field configuration A^i, one can always calculate its winding number from a volume integral which in the temporal gauge is given by : n = \frac\int d^3 r \ \text(\epsilon_A^iA^jA^k), where g is the coupling constant. The different classes of vacuum states with different winding numbers , n\rangle are referred to as topological vacua.


Theta vacua

Topological vacua are not candidate vacuum states of Yang–Mills theories since they are not eigenstates of large gauge transformations and so aren't gauge invariant. Instead acting on the state , n\rangle with a large gauge transformation \Omega_ with winding number m will map it to a different topological vacuum \Omega_m, n\rangle = , n+m\rangle. The true vacuum has to be an eigenstate of both small and large gauge transformations. Similarly to the form that eigenstates take in periodic potentials according to
Bloch's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
, the vacuum state is a coherent sum of topological vacua : , \theta\rangle = \sum_n e^, n\rangle. This set of states indexed by the angular variable \theta \in [0,2\pi) are known as ''θ''-vacua. They are eigenstates of both types of gauge transformations since now \Omega_m, \theta\rangle = e^, \theta\rangle. In pure Yang–Mills, each value of \theta will give a different ground state on which excited states are built, leading to different physics. In other words, the Hilbert space decomposes into superselection, superselection sectors since expectation values of gauge invariant operators between two different ''θ''-vacua vanish \langle \theta, \mathcal O , \theta' \rangle = 0 if \theta \neq \theta'. Yang–Mills theories exhibit finite action solutions to their equations of motion called
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
s. They are responsible for tunnelling between different topological vacua with an instanton with winding number \nu being responsible for a tunnelling from a topological vacuum , n_-\rangle to , n_+\rangle = , n_-+\nu\rangle. Instantons with \nu=\pm 1 are known as
BPST instanton In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU(2) Yang–Mills the ...
s. Without any tunnelling the different ''θ''-vacua would be
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
, however instantons lift the degeneracy, making the various different ''θ''-vacua physically distinct from each other. The ground state energy of the different vacua is split to take the form E(\theta) \propto \cos \theta, where the constant of proportionality will depend on how strong the instanton tunnelling is. The complicated structure of the ''θ''-vacuum can be directly incorporated into the Yang–Mills
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
by considering the vacuum-vacuum transitions in the path integral formalism : \lim_\langle \theta, e^, \theta\rangle = \int \mathcal D A e^. Here H is the Hamiltonian, S the Yang–Mills action, and \mathcal L_\theta is a new CP violating contribution to the Lagrangian called the ''θ''-term : \mathcal L_\theta =\theta \frac\text ^\tilde F_ where \tilde F^ = \tfrac\epsilon^F_ is the dual field strength tensor and the trace is over the group generators. This term is a total derivative meaning that it can be written in the form \mathcal L_\theta = \partial_\mu K^\mu. In contrast to other total derivatives that can be added to the Lagrangian, this one has physical consequences in non-perturbative physics because K^\mu is not gauge invariant. In quantum chromodynamics the presence of this term leads to the strong CP problem since it gives rise to a neutron electric dipole moment which has not yet been observed, requiring the fine tuning of \theta to be very small.


Modification due to fermions

If massless fermions are present in the theory then the vacuum angle becomes unobservable because the fermions suppress the instanton tunnelling between topological vacua. This can be seen by considering a Yang–Mills theory with a single massless fermion \psi(x). In the path integral formalism the tunnelling by an instanton between two topological vacua takes the form : \begin \langle n, n+\nu\rangle & \sim \int \mathcal D A \mathcal D \psi \mathcal D \bar \psi \exp\bigg(-\int d^4 x \frac\text F^F_+i\bar \psi \psi\bigg) \\ & \sim \int \mathcal D A \det (i) \exp\bigg(-\int d^4x \frac\text F^F_\bigg). \end This differs from the pure Yang–Mills result by the fermion determinant acquired after integrating over the fermionic fields. The determinant vanishes because the Dirac operator with massless fermions has at least one zero eigenvalue for any instanton configuration. While instantons no longer contribute to tunnelling between topological vacua, they instead play a role in violating axial charge and thus give rise to the chiral condensate. If instead the theory has very light fermions then the ''θ''-term is still present, but its effects are heavily suppressed as they must be proportional to the fermion masses.


See also

*
Instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
* Strong CP problem


References

{{Reflist Gauge theories Quantum chromodynamics