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A field in physics can be imagined as something like mattress. At every point in space we have an harmonic oscillator and these oscillators are connected. (See the laymen explanation of Quantum Field Theory). In this picture, the vacuum state is when all these oscillators sit still. In a quantum field theory, this is never possible because there are always vacuum oscillation. Particles are then excitations of these ground state mattress.
This picture is quite good but misses one crucial thing. There can be tunneling processes in the vacuum. The thing is that if we study the vacuum carefully, we notice that the vacuum state is not so simple as a set of harmonic oscillators that move a little. Instead much better approximation is a set of connected pendulums. At each point in space, we have a pendulum instead of an oscillator and these pendulums are connected.
This is an important distinction, because in a quantum theory a pendulum can do something in addition to the usual small vacuum oscillation around its minimum. The pendulum can tunnel such that moves once around its revelation and ends up again in its ground position. While such a process would require usually a lot of energy, in a quantum theory it can happen even at minimum energy, i.e. when the quantum pendulum is in a ground state. This is known as a tunneling process. In this specific context, the tunneling is also called an instanton process.
These processes are important, because we need to take them into account when we do calculation in quantum field theory. All particles are exictations above the ground state and describe the ground state properly, we need to take these tunneling processes into account.
If we use the standard perturbative form for the fields and vacuum state then $\langle \bar \Psi \Psi \rangle = 0$ and $\langle G_{\mu \nu}^a G_{\mu \nu}^a \rangle = 0 $ etc. So we might think these terms do not even exist. However, we know from chiral symmetry breaking, i.e. from the fact that $m_\rho\neq m_{a_1}$ and that the pion exists, that expressions like $\langle \bar \Psi \Psi \rangle \equiv \langle \text{true. vac. } | \bar \Psi \Psi |\text{true. vac. } \rangle \neq 0$. We cannot calculate these matrix elements. They reflect the deepest non-perturbative aspects of the theory.
But we can calculate the $k^2$ dependence of the coefficients of these operators in the operator product expansion. Thus we can produce a QCD calculation of $\Pi(k^2)$ [call it $\Pi^{QCD}(k^2)$] which includes the perturbative term plus corrections to it that involve powers of $(k^2)^{-d/2}$ multiplied by unknown numbers, the unknown values of the vacuum matrix elements. […] Schematically then $$ \Pi^{QCD}(k^2) = \Pi^{pert.}(k^2)+a \frac{\langle G_{\mu \nu}^a G_{\mu \nu}^a \rangle}{k^4} + b \frac{ \langle \bar \Psi \Psi \rangle }{k^4} + \ldots $$ where $a$ and $b$ are known. page 303 in An Introduction to Gauge Theories and Modern Particle Physics, Vol 2 by Elliot Leader,Enrico Predazzi
Analogies to the QCD Vacuum:
Another simple model, the Lagrangian for a pendulum, that shows a similar behaviour is described in The Interpretation of Pseudoparticles in Physical Gauges by Claude W. Bernard, Erick J. Weinberg.
The Lagrangian is
$$ \mathcal{L} = \frac{1}{2} \dot{x}^2+ \lambda cos(x), $$
where, without any physical input, we use $ -\infty < x < \infty$. Then, this Lagrangian describes a particle in a periodic potential, similar to the Bloch wave example, well known from solid state physics. The energy spectrum is a series of "bands" and the states fall in sectors, labelled by some parameter $\beta$. (In solid state physics, the parameter $\beta$ is the phase that the wave function picks up when it tunnels from one ground state to another one.)
However, we can also view this Lagrangian as the Lagrangian for a pendulum. In this case, the parameter $x$ becomes an angular variable that denotes to position of the pendulum. Therefore: $0 \leq x \leq 2 \pi$. This means directly that we now identify $x$ and $x + 2\pi$. Now, we don't get a band structure, but instead discrete energy levels. However, we are free to add a term $b \dot x$ to the Lagrangian. This term is a total derivative and therefore has no influence on the classical equations of motion. Nevertheless, it will shift the energy levels in the quantum theory. Each value for $b$ leads to a spectrum that is equivalent to a particular sector in the periodic potential.
They also note that instead of adding the term to the Lagrangian, we can impose periodic boundary conditions: $\psi(0) = e^{ia} \psi (2\pi)$, instead of $\psi(0) = \psi (2\pi)$ and the result would be exactly the same.
First, they note that the temporal gauge is convenient for the description of the QCD vacuum, because it allows us to rewrite the usual free Lagrangian:
$$ \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} $$
in a form that resembles a non-relativistic potential problem:
$$ L = \int dx \frac{1}{2} (\partial_0 A_i)^2 -V(A), $$
where $V= \int dx \frac{1}{4} F_{\mu\nu} F^{\mu\nu} $.
Then they consider an analogous quantum mechanical example with $H = \frac{1}{2} M\left( \frac{dx}{dt}\right)^2+\frac{\lambda}{r^2}$. The potential $V= \frac{\lambda}{r^2}$ is infinite at $r=0$. Hence the subspace of $\mathbb{R}^2$ that correspond to finite potential values is the normal euclidean space with the point zero removed: $\mathbb{R}^2 / \{ 0 \} \sim S^2$. Therefore, this subspace is, in contrast to the full space $\mathbb{R}^2$ not simply connected.
A more convenient parametrization for this problem is given by polar coordinates: $r= (x_1^2+x_2^2)^{1/2}$ and $\phi = \arctan(x_2/x_1)$ with $0 \leq \phi \leq 2\pi$.
The set of all paths in the subspace $\mathbb{R}^2 / \{ 0 \}$ can be classified by the integral over the derivative of the angular variable $\phi$:
$$ \oint d\tau \frac{1}{r^2} \epsilon_{\alpha \beta}x_\alpha \dot{x}_\beta \equiv n , $$ where $\dot{x}_\beta= dx_\beta/d\tau$. (This is completely how we usually measure the length of a path. The length of a path is given by the integral over the derivative of the position vector. Here, we only count how much "angular length" a given path has.)
If we consider a path that winds multiple times around the hole at $r=0$, we notice that the variable $\phi$ is not continuous along such a path. The reason is the jump at $\phi = 2\pi$, where as the path goes on $\phi$ jumps to $0$ (because we defined it, as usual, to be on the interval $ [0,2\pi]$). This is completely analogous to the "gauge jump" that can be observed for the QCD vacuum in the Coulomb gauge, i.e. to the famous Gribov ambiguities.
To avoid such a discontinuity, we can enlarge the range of definition for $\phi$. If we allow that $\phi$ takes values all along the real line, i.e. $-\infty \leq \phi \leq \infty$, there is no longer a discontinuity. (Mathematically this means we now consider the covering space). Physically they call this process "unwinding" the angular variable.
In addition, they use their quantum mechanical analogue to motivate why we only consider those gauge transformations that satisfy
$$ g(x) \to 1 \quad \text{ for } |x| \to \infty.$$
They argue:
On the other hand, if we also include those paths connecting an initial field colrfiguration A(x) with a gauge transform Ae(x) thereof with G(x) violating the above-mentioned boundary condition (such paths would necessarily imply that at least for some interval along the path the potentials are not pure gauge at spacial infinity), then integral (14) is no longer a topological invariant, ~nd the representation of the canonica1 momentum given by eq. (31) becomes singular at spacial i~ffinity because of the surface term appearing in (13) (which in general will no longer vanish). ~ut in this case the physical configuration space would be simply connected, and all representations of the canonical momentum given by eqs. (22) and {24) with ~qS[A]/3Aa(x) well defined everywhere are equivalent. As has been pointed out in ref. (7), however, field configurations approaching a pure gauge at spacial infinity with the gauge function G(x) violating condition (8) are separated from the ones we have included by an infinite potential barrier, and presumably there exists no tunnelling between them; hence it was consistent to restrict ourselves to those gauge transformations satisfying eq. (8).
We close this paper by illustrating some of the above remarks in a simple idealized quantum-mechanical model in two space dimensions labelled by the co-ordinates (x~ x2). Thus consider the motion of a particle of mass M in a potential V® depending only on the radial co-ordinate r --~ V~, z + x~, which is infinite within a disc of radius R. ]n this ease the wave function vanishes identically for r <~R, and the physically relevant configuration space S consists of all points outside the disc where V® is finite. Thus all continuous paths in S can be classified by a winding number given by er (16). On the other hand, including also paths for which the energy is unbounded, the configuration space becomes clearly simply connected. Nevertheless, as far as the quantum dynamics is concerned, the region of the disc is irrelevant, and the circle r = R represents effectively a boundary of the physical configuration space where wave functions have to vanish. Now, as a consequence of the multiply connected structure of the physical configuration space, the theory will exhibit a quantization ambiguity, since …
Comments About Theta Vacua and Topology of the SU(2) {Yang-Mills} Theory H.O. Girotti, H.J. Rothe
Interpretations of the QCD Vacuum
For this kind of interpretation, see also The Interpretation of Pseudoparticles in Physical Gauges by Claude W. Bernard, Erick J. Weinberg and page 277 in Rubakov's "Classical Theory of Gauge Fields".
The Yang-Mill vacuum is a Bloch wave
Quantum Meaning of classical field theories by R. Jackiw
This is the standard interpretation. Nice, detailed discussions can be found in
Moreover, in THE ROLE OF SURFACE VARIABLES IN THE VACUUM STRUCTURE OF YANG-MILLS THEORY S. WADIA and T. YONEYA the authors argue that "Vacuum periodicity is not an artifact of the A 0 = 0 gauge […] In other gauges the additional dynamical variables on the surface at spatial infinity play an essential role in describing the vacuum structure of Yang-Mills field. "
The interpretation of the QCD vacuum is highly gauge dependent.
Usually one invokes the temporal gauge, which leaves a residual gauge freedom and leads to the usual periodic vacuum picture.
In contrast, in a physical gauge like the axial gauge there is only one unique classical ground state: $A_\mu =0$:
in nonAbelian theories this can be done only at the (aesthetic) cost of allowing finite energy configurations for which the potentials do not vanish at spatial infinity. In other gauges (e.g. temporal gauge, $A_0=0$) these configurations are avoided, but an infinite set of degenerate classical ground states remains (92, 93). Tunneling also occurs in gauges with unique classical vacua (94). In these gauges the instanton describes a process in which the field starts at the vacuum and then tunnels through a potential energy barrier simply to get back to where it started. This is somewhat analogous to the case of a particle constrained to move on a vertically oriented ring, with the energy of the particle being too small to overcome the gravitational potential energy barrier at the top of the ring. Classically the particle will stay at the bottom of the ring, but quantum mechanically it can go around the ring by tunneling. Although its description is rather different in these two classes of gauges, the observable consequences of this instanton-induced tunneling are, of course, the same in all gauges (95, 96) .
The QCD vacuum in the axial gauge is discussed, for example, in The Interpretation of Pseudoparticles in Physical Gauges by Claude W. Bernard, Erick J. Weinberg. They write:
In the $A_0 = 0$ gauge, however, one must decide whether "configuration'" in this prescription is to be defined by the $A_i$, or by the $F_{ij}$. The former is the choice made by CDG and JR; however, this method of parametrizing function space necessitates the introduction of many vacuums.[…] The possibility of two different parametrizations occurs because in the $A_0 = 0 $gauge the $F_{\mu \nu}$ do not uniquely determine the canonical variables; […] There is no observable distinction between the two descriptions since the physical gauge Lagrangian can be adjusted to correspond to any desired sector, in particular to the sector corresponding to the observed universe. Thus, the many vacuums are not essential to the understanding of pseudoparticles; rather, they are artifacts of a particular way of parametrizing the function space and thus of defining the path integral. It is quite possible, and in some gauges necessary, to do without them.
The Interpretation of Pseudoparticles in Physical Gauges by Claude W. Bernard, Erick J. Weinberg
However, in THE ROLE OF SURFACE VARIABLES IN THE VACUUM STRUCTURE OF YANG-MILLS THEORY S. WADIA and T. YONEYA the authors argue that "Vacuum periodicity is not an artifact of the $A_0 = 0$ gauge […] In other gauges the additional dynamical variables on the surface at spatial infinity play an essential role in describing the vacuum structure of Yang-Mills field. "
Even stranger is the interpretation in the Coulomb gauge. In this gauge one encounters the famous Gribov ambiguities:
the BPST instanton, truly a gauge-invariant object, tunnels between the two Gribov vacua in the Coulomb gauge [18], and, allowing discontinuities in the time evolution of the gauge fields, there seems to be a rich tunneling picture [19-21].
ON THE VACUUM STRUCTURE IN THE COULOMB AND LANDAU GAUGES by Antti NIEMI
However, it is important to note that the existence of Gribov copies and the usual periodic vacuum picture depend crucially on the compactification of Euclidean space $R^4$ to $S^4$. This compactification is achieved by restricting the discussion to only those gauge transformations which satisfy $g(x) \to 1$ as $|x| \to \infty$. (The result is the same when they don't go to $1$ but to some other constant matrix.) The crucial thing here is that we assume that infinity is only a point and that the gauge transformations do not go to different matrices, when we approach $|x| \to \infty$ from different directions. This means, we assume there is no angular dependence at $|x|$. The usual homotopy classification is only possible, because through the restriction to only those gauge transformations that satisfy $g(x) \to 1$, the configuration space is no longer simply connected.
However, there seems to be no good reason for this requirement and it is only used to get to the nice periodic picture for the vacuum.
"there is actually no very convincing argument to justify this restriction"
Quantum Field Theory by Claude Itzykson,Jean-Bernard Zuber
The main difficulty in the barrier-penetration description of the $\theta$ vacua is that in order to show that the initial and final configurations are in different topological sectors, one must compactify the field configurations. This is done by imposing a condition on the configurations which is stronger than the assumption of finite action. The justification of this compactification has always been recognized as weak but it had seemed necessary.
However, it is also possible to achieve a homotopical classification of the vacuum without this restriction as shown in Nontrivial homotopy and tunneling by topological instantons. Moreover, the physics does not change in an axial gauge where there is no such requirement (and no periodic potential), as shown in The Interpretation of Pseudoparticles in Physical Gauges by Claude W. Bernard, Erick J. Weinberg.
There is a non-zero energy density of the QCD vacuum due to non-perturbative effects (source: Eq. 1.26 |here)
$$ \epsilon_{vac} \simeq -\frac{b}{128\pi^2}\langle 0|(gG_{\mu\nu}^a)^2|0\rangle \simeq 0.5 \mathrm{\ GeV/fm^3}$$
This contribution is negative, which means that non-perturbative effects lower the vacuum density we get if we only consider perturbative effects. The absolute value of $\epsilon_{vac} $ sets a scale for the energy density that is necessary to rearrange the vacuum structure.
It is not known completely what is really going in the QCD vacuum, because the final solution of the QCD equations has not been found yet. The values for the vacuum energy density are inferred, for example, by fitting QCD sum rules to experimental data.
To think is difficult. To think about nothing is more difficult than about something. Lev Okun
Consider the theory of instantons and remember that standard QCD theory says that the vacuum which we inhabit is an instanton sea with about one instanton per femtometer. This means that the physical reality which we inhabit, if you remove everything and just consider the plain vacuum, is already densely filled with, if you wish, physical incarnation of identity types.
The main point is that in order to describe tunneling effects semiclassical, we need to work in Euclidean spacetime, and hence we Wick rotate: $t \to i \tau$.
This, however implies $A_0 \to i A_0$, because the $F_{0i}^a$ components of the field strength tensor become under Wick rotations:
$$ F_{0i}^a = \partial_0 A_i^a - \partial_i A_0^a+ g f^{abc} A_0^b A_i^c \to i \partial_0 A_i^a - \partial_i A_0^a+ g f^{abc} A_0^b A_i^c . $$
The reason for the appearance of $i$ here, is that from $t \to - i \tau$ it follows that $\partial_0 = \frac{\partial}{\partial t} \to \frac{\partial}{\partial i \tau} = -i \frac{\partial}{\partial\tau} = -\partial_0$.
Without $A_0 \to i A_0$, the components $F_{0i}^a$ are complex and this means immediately that our Euclidean action and the field equations are complex. This is not what we want to describe tunneling phenomena.
However, if we substitute $A_0 \to i A_0$ at the same time as $t \to i \tau$, the $F_{0i}^a$ are purely imaginary and the Euclidean action is real:
$$ S = \int d^3x dt \left[ \frac{1}{2} F_{0i}^a F_{0i}^a - \frac{1}{4} (F_{ij}^a)^2 \right] \to i S_E , $$
where $S_E= \int d^3x dt\frac{1}{4} (F_{ij}^a)^2 $.
The temporal gauge
$$ A_0^a = 0$$
is useful, because in this gauge the action of the gauge fields in Minkowski spacetime
$$ S_M= \int d^4 x \left[ \frac{1}{2} (\partial_0 A_i^0)^2 - \frac{1}{4} (F_{ij}^a)^2 \right] $$
yields the corresponding action in Euclidean spacetime simply via the usual rule $t \to - i \tau$
$$ S_E= \int d^4 x \left[ \frac{1}{2} (\partial_\tau A_i^0)^2 + \frac{1}{4} (F_{ij}^a)^2 \right] $$
and thus is still in the "canonical form".
Anatoly Larkin, posed a challenge to two outstanding undergraduate teenage theorists, Sacha Polyakov and Sasha Migdal: “In field theory the vacuum is like a substance; what happens there?”Chapter 9 in The Infinity Puzzle, by F. Close
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