2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationspacetime and Lorentz invariance

Special relativity and spacetime — preserving a quadratic form with one sign different narrows down the form of the action

We write the wave equation built in 2-02 with one dimension of space and wave speed $c$.

In Newtonian mechanics the coordinates of an observer moving to the right with speed $v$ are $t'=t,\ x'=x-vt$ (a Galilean transformation). Substituting $\partial_t=\partial_{t'}-v\partial_{x'}$, $\partial_x=\partial_{x'}$, the operator of (1) changes form.

Example. Substituting $u=f(x'-st')$ gives $s^2+2vs-(c^2-v^2)=0$, that is, $s=c-v$ and $-(c+v)$. For $v=c/2$ the waves look like $0.5c$ to the right and $1.5c$ to the left Computed.

For sound this is correct, because the observer at rest relative to the air is special. But the wave equation of light comes from Maxwell's equations (2-03) and needs no medium. Einstein (1905) took as premises that the laws have the same form in every inertial frame and that the speed of light does not depend on the motion of the source Known. We turn the question round. Which transformations do not change the form of (1)?

For linear transformations, preserving the operator (1) as it is is equivalent to preserving the following interval (the two coefficient matrices change in mutually inverse ways; scaling up or down preserves equation (1) but not the interval).

Rotations preserve $x^2+y^2$ and could be written as $\cos,\sin$. The transformations that preserve $s^2$ can be written, with $\cosh^2\varphi-\sinh^2\varphi=1$, as $\cosh,\sinh$.

We call $\varphi$ the rapidity ($\tanh\varphi=v/c$), and this transformation a boost. Putting $\gamma=\cosh\varphi=1/\sqrt{1-v^2/c^2}$ gives $t'=\gamma(t-vx/c^2),\ x'=\gamma(x-vt)$, and for $c\to\infty$ we return to the Galilean transformation (§3 of Einstein 1905 has the same form, except that the speed of light is written $V$) Known. The difference from rotations is in §01 and §04 of the spacetime article. Example. For $v=0.5c$, $\gamma=1.1547$. The time $t$ of a clock at $x=0$ is $t'=\gamma t$ for the moving observer, and each tick looks longer.

ComputedThe numbers in the figure are computed on the spot from (3). Pressing a row of Table 1 (or selecting it with Tab and pressing Enter) sets the figure to that speed.

ComputedTable 1 $\gamma$, rapidity, and the ratio of kinetic energies ($(\gamma-1)mc^2$ ÷ $mv^2/2$). $\gamma=\cosh\varphi$ is cross-checked.

The $s^2$ of the difference between two events does not change under boosts, and its sign is the same for whoever looks.

Two timelike-separated points can be connected by a signal slower than light, and under transformations that do not reverse the direction of time (section 04) their order is the same for whoever looks. Two spacelike-separated points cannot be connected by any signal, and their order is exchanged depending on the observer. The view of time and space as a single four-dimensional geometry became widely known through Minkowski's lecture of 1908. The treatment that introduces an imaginary time coordinate $t\sqrt{-1}$ and sees Lorentz transformations as rotations in four dimensions is already in Poincaré (1906) Known.

Example (Figure 1). $c=1$. The origin O and B $=(t,x)=(0.4,1.3)$ are spacelike-separated, with $s^2=1.53\gt0$. $t'_B=\gamma(0.4-1.3v)$ becomes negative at $v\gt0.308$ ($\varphi\gt0.318$), and B moves from "after" O to "before" it. C $=(1.5,0.5)$ is timelike-separated, with $s^2=-2$, and stays $t'_C=\gamma(1.5-0.5v)\gt0$.

If two points whose order is exchanged were causally connected, there would be a contradiction. In quantum field theory this becomes the axiom "field operators at spacelike-separated points commute" (Jaffe–Witten §3, 4-09). For the light cone and "the degeneration of distance", see §03 of the spacetime article.

We return space to three dimensions and write $x^\mu=(ct,x,y,z)$. When the same index appears once up and once down, it is summed.

To lower an index we multiply by $\eta$ ($x_\mu=(-ct,x,y,z)$). $\partial_\mu=\partial/\partial x^\mu$, $\Box=\eta^{\mu\nu}\partial_\mu\partial_\nu=-\partial_t^2/c^2+\nabla^2$. The signature $(-,+,+,+)$ is the same as in the spacetime article; some textbooks use the opposite convention.

A Lorentz transformation is a $4\times4$ real matrix $\Lambda$ satisfying the right-hand formula of (4), and these form a group $\mathrm O(1,3)$ (1-02). $\mathrm O(1,3)$ splits into four parts according to $\det\Lambda=\pm1$ and the direction of time (whether the top-left component is $\ge1$ or $\le-1$; by (4) its square is at least 1). The part containing the identity matrix, $\mathrm{SO}^+(1,3)$ ($\det\Lambda=1$, preserving the direction of time), is generated by 3 rotations and boosts in 3 directions, and the group obtained by adding the 4 translations is the Poincaré group Known. The quantum field of the problem statement is placed on a Hilbert space carrying a representation of this group (Jaffe–Witten §3).

Example. $\mathrm{SO}(4)$ is compact, but $\mathrm{SO}^+(1,3)$ is not, since $\varphi$ can be taken as large as we like. The "compact simple gauge group" of the problem statement concerns the internal symmetry (1-04); the Lorentz group of spacetime is non-compact from the start.

We call the time of a clock carried by the particle its proper time $\tau$. From the invariance of $s^2$, $c^2d\tau^2=c^2dt^2-|d\vec x|^2$, that is, $d\tau=dt/\gamma$. Since $\tau$ is the same for everyone, the four-momentum $p^\mu=m\,dx^\mu/d\tau$ changes by $\Lambda$, just as $x^\mu$ does.

The right-hand formula is $E^2=|\vec p|^2c^2+m^2c^4$. Although $E$ and $\vec p$ differ from observer to observer, mass is the invariant length of the four-momentum. Example. $E=\gamma mc^2=mc^2+\tfrac12mv^2+\tfrac38mv^4/c^2+\cdots$, and the rightmost column of Table 1 is this correction.

Why it is called the "mass" gap

The dispersion relation $\omega^2=c^2k^2+\Omega^2$ of 2-02, read as $E=\hbar\omega,\ p=\hbar k$ (2-05), is (5) itself, with $\Omega=mc^2/\hbar$. The problem statement defines the mass gap as "the spectrum of the Hamiltonian $H$ (the set of values the energy can take; defined in 2-05 and 1-10) has no points in $(0,\Delta)$" (Jaffe–Witten §4). When three things hold together, Lorentz invariance, positive energy ($H\ge0$ holds for every observer) and a unique vacuum (Jaffe–Witten §3), this is the same as "the pairs of energy and momentum $(E,\vec p)$ of states other than the vacuum satisfy $E\ge\sqrt{|\vec p|^2c^2+\Delta^2}$". For if a pair violates it, then if it is spacelike ($E\lt|\vec p|c$) a boost makes the energy negative, contradicting positive energy; if it is timelike, in its rest frame, and if it is lightlike, by a boost that makes it as small as we like, it enters $(0,\Delta)$. That only the vacuum has the pair $(0,\vec 0)$ uses the premise that the vacuum is unique Known. $\Delta$ can be read as the $mc^2$ of the lightest state (4-11).

If the action (2-01) is the same for everyone, so is the form of the equation of motion. The simplest invariant of a world line is the proper time, so

The coefficient was chosen so that the slow-motion expansion $L=-mc^2+\tfrac12m|\vec v|^2+\cdots$ matches Newton. The equation of motion is $\frac{d}{dt}(\gamma m\vec v)=0$, so the world line is straight, and the Noether conserved quantity of time translation (2-01) is $E=\gamma mc^2$. From the sign, minimising the action and maximising the proper time are the same thing, and among timelike world lines with fixed endpoints the straight line maximises the proper time. Seen in the frame of an observer for whom both ends are at the same place (such a frame exists because the ends are timelike-separated; call the time difference there $T$), $d\tau=\sqrt{1-|\vec v|^2/c^2}\,dt\le dt$, with equality only when $\vec v\equiv0$. Hence $\tau\le T$, and the maximum is attained only by the straight line ($\tau$ is the same in every frame). We call this the reverse triangle inequality Known. A stationary point need not be a minimum (2-01), but here it really is a minimum, and the twin paradox is a restatement of this.

Example. The proper time of $T=1$, $x(t)=(u/\pi)\sin\pi t$ (top speed $u$) is $\tau=\tfrac2\pi\mathcal E(u)$ with $\mathcal E(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2\theta}\,d\theta$ (the complete elliptic integral of the second kind). For $u\to1$, $2/\pi\approx0.637$.

ComputedThe numbers in the figure are computed on the spot by the trapezoidal rule for $d\tau=dt/\gamma$ (400 subdivisions).

ComputedTable 2 $\tau/T$ of world lines with fixed endpoints. For the curve, the numerical integral and $\tfrac2\pi\mathcal E(u)$ agree to $2\times10^{-16}$. The broken line is the path that goes out and back at speed $u$ (the twins).

The action of a field is $S=\frac1c\int\mathcal L\,d^4x$. Since $d^4x$ does not change under $|\det\Lambda|=1$, the action is invariant if the Lagrangian density $\mathcal L$ is a scalar. If we also impose the condition of being local (only the field at that point and its first derivatives), the form of the terms is strongly narrowed down.

For a scalar field, $\partial_\mu\phi$ is contracted with $\eta$.

The right-hand one is the Klein–Gordon equation. Example. In one dimension, multiplying the whole by the tension $\sigma$ gives the string with a mass term of 2-02, with the correspondence $c^2=\sigma/\rho$, $\Omega=c\mu$.

The electromagnetic field. Gather the potentials $\phi,\vec A$ of 2-03 (this $\phi$ is different from the field of (7)) as $A_\mu=(-\phi/c,\vec A)$, and put $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. The components are $F_{0i}=-E_i/c$, $F_{ij}=\varepsilon_{ijk}B_k$ ($\varepsilon_{ijk}$ is the symbol with $\varepsilon_{123}=1$ that changes sign when indices are exchanged), and it gathers the electric and magnetic fields into one antisymmetric matrix. It does not change under the gauge transformation $A_\mu\to A_\mu+\partial_\mu\chi$ ((G) of 2-03). Since $F^{0i}=-F_{0i}$, $F^{ij}=F_{ij}$, we have $F_{\mu\nu}F^{\mu\nu}=-2|\vec E|^2/c^2+2|\vec B|^2$, and

In SI units the whole is multiplied by $1/\mu_0$, and this becomes the part of (L) of 2-03 without $\rho,\vec J$. The equation of motion $\partial_\mu F^{\mu\nu}=0$ is half of Maxwell's equations in vacuum, and the other half holds as soon as $F$ is built from $A$.

Yang–Mills fields. Make $A_\mu=A^a_\mu T^a$ a field with values in a Lie algebra (1-03, 1-04), and add a term with the structure constants $f^{abc}$ to $F$ (3-02).

In matrices, $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu]$ ($T^a$ Hermitian with $[T^a,T^b]=if^{abc}T^c$, the convention of 1-03; $g$ absorbed into $A$). The last term is the only difference from Maxwell, and the field becomes its own source. Even so, the classical equations describe "massless waves travelling at the speed of light" (Jaffe–Witten §1) Known.

If we also impose gauge invariance (3-02) and the group is simple (the invariant quadratic form is unique up to a constant multiple; 1-03), the Lorentz-invariant quadratic expressions in $F$ are only the linear combinations of $\sum_aF^a_{\mu\nu}F^{a\,\mu\nu}$ and $\sum_a\varepsilon^{\mu\nu\rho\sigma}F^a_{\mu\nu}F^a_{\rho\sigma}$ (the four-dimensional version of the symbol $\varepsilon^{0123}=1$) Known. For the electromagnetic field the second is proportional to $\vec E\cdot\vec B$; it changes sign under spatial reflection and, being a total derivative, does not change the equations of motion (the $\theta$ term of 3-05). Without the condition, the $\sum c_{ab}F^a_{\mu\nu}F^{b\,\mu\nu}$ built from an arbitrary symmetric matrix $c_{ab}$ is also invariant.

With $c=1$, length and time are put on the same footing, and with $\hbar=1$ energy and inverse time, so one unit remains. If mass has dimension 1, length has $-1$. The only conversion is $\hbar c=197.3269804\ldots\ \mathrm{MeV\,fm}$ ($1\,\mathrm{fm}=10^{-15}\,\mathrm m$). The gap $\Delta$ corresponds to a length $\hbar c/\Delta$, a rough measure of how far correlations reach. If there is a gap, then for a local quantity $O$ whose vacuum expectation value $\langle O\rangle$ is 0, the correlation $|\langle O(\vec x)O(\vec y)\rangle|$ at two points at the same time and far enough apart is at most $\exp(-C|\vec x-\vec y|)$ for any $C$ satisfying $C\lt\Delta/(\hbar c)$ (Jaffe–Witten §5; for $\hbar=c=1$, $C\lt\Delta$). If the expectation value is not 0, the correlation approaches $\langle O\rangle^2$.

ComputedTable 3 Converting masses into lengths ($\hbar c$ computed from the defined SI values $h,c,e$).

The action is dimensionless (measured in $\hbar$), so in a spacetime of dimension $d$, $[\mathcal L]=d$, $[\partial]=1$. Since $(\partial\phi)^2$ and $\lambda\phi^4$ have dimension $d$, and since $F$ of (9) is the sum of $\partial A$ and $AA$ with $[A]=1$, $[F^2]=4$,

ComputedTable 4 Mass dimensions ($\hbar=c=1$, computed with fractions).

In four dimensions, the action (9) contains not a single constant with dimensions. Neither a mass nor a length can be built from the coefficients, and classical Yang–Mills theory has no scale Known. If in the quantum theory there is $\Delta\gt0$, the scale must come from outside the classical action; in physics it is thought to enter in the process of renormalisation, and this is called dimensional transmutation (4-07) Physics. The exposition too gives the fact that in four-dimensional gauge theory the mass gap is not classically visible ('the mass gap is not classically visible') as a reason why a non-classical change of variables may be needed (Jaffe–Witten §6.6). In two and three dimensions $g^2$ itself is a scale (4-04, 6-06). For $\phi^4$ see 6-08; for other ways four dimensions is special, see the four-dimensions article. A figure that moves the $d$ of Table 4 continuously and sets it beside other quantities that switch at four is the dimension dial of 1-12 §10.

Substituting $t=-i\tau$ ($\tau$ real) into (2) gives $-c^2t^2=c^2\tau^2$, and the interval becomes the square of the Euclidean distance $c^2\tau^2+|\vec x|^2$. This is the Wick rotation (the name is said to come from Wick 1954), which formally puts back the difference of sign; it is the same operation as the $ict$ of §02 of the spacetime article (the authoritative treatment of its form for actions and correlations of fields is 4-02 §02). Since $\partial_t^2=-\partial_\tau^2$, $\Box$ becomes the four-dimensional Laplacian, (7) becomes $(-\partial_\tau^2/c^2-\nabla^2+\mu^2)\phi=0$, and oscillation turns into decay. The one-dimensional version, dropping the dependence on space and putting $c=1$, is (11). $\delta(\tau)$ is the thing concentrated at $\tau=0$ with $\int f(\tau)\delta(\tau)\,d\tau=f(0)$ (strictly a distribution rather than a function; 1-11).

Example. For $\tau\ne0$, $G''=\mu^2G$; at $\tau=0$ the slope $G'$ jumps by $-1$, so $G''$ contains $-\delta$, and $-G''+\mu^2G=\delta$. In three dimensions, $e^{-\mu r}/(4\pi r)$ Known. On the Euclidean side, mass appears as a rate of decay (an inverse length). This is the starting point of 4-11, which ties the mass gap to the correlation length. Writing the Euclidean theory as a probability measure (4-02) and the conditions for reconstructing the Minkowski side (Osterwalder–Schrader) are the subject of 4-10. This chapter has only shown the substitution.

What this chapter can say

Sources and reproduction

Next chapter: 2-05 The bones of quantum mechanics. Imaginary time continues in 4-02, and spacetime itself in One sign is different.

← 2-03 Electromagnetism is a gauge theoryContents2-05 The bones of quantum mechanics →