[Weinberg QFT] 2.3 Quantum Lorentz Transformations

 This article is one of the posts in the Textbook Commentary Project.

Lorentz Transformation

Einstein's principle of relativity states the equivalence of certain 'inertial' framse of reference. It is distinguished from the Galiliean principle of relativity, obeyed by Newtonian mechanics, by the transformation connecting coordinate systems in different inertial framse. If $x^\mu$ are the coordinates in one inertial frame (with $x^1,x^2,x^3$ Cartesian space coordinates, and $x^0=t$ a time coordinates, the speed of light being set equal to unity) the in any other inertial frame, the coordinates $x'^\mu$ must satisfy $$\begin{align}\eta_{\mu\nu} dx'^\mu dx'^\nu = \eta_{\mu\nu} dx^\mu dx^\nu\end{align}$$ or equivalently \begin{align}\eta_{\mu\nu} \frac{\partial x'^\mu}{\partial x^\rho} \frac{\partial x'^\nu}{\partial x^\sigma} = \eta_{\rho \sigma}\end{align}$$

Here $\eta_{\mu\nu}$ is the diagonal matrix, with elements $$\begin{align}\eta_{11} = \eta_{22} = \eta_{33} = +1,\ \eta_{00} = -1\end{align}$$ and the summation convention is in force: we sum over any index like $\mu$ nad $\nu$ in Eq. (2), which appears twice in the same term, once upstairs and once downstairs. These transformations have the special property that the speed of light is the same (in our units, equal to unity) in all inertial frames; a light wave traveling at unit speed satisfies $\left| d\mathbf{x}/dt\right|=1$, or in other words $\eta_{\mu\nu}dx^\mu dx^\nu=d\mathbf{x}^2-dt^2=0$, from which it follows that also $\eta_{\mu\nu} dx'^\mu dx'^\nu=0$, and hence $\left| d\mathbf{x}'/dt'\right|=1$.

Any coordinate transformation $x^\mu\rightarrow x'^\mu$ that satisfies Eq. (2) is linear $$\begin{align}x'^\mu = \Lambda^\mu_\nu x^\nu + a^\mu\end{align}$$ with $a^\mu$ arbitrary constants, and $\Lambda^\mu_\nu$ a constant matrix satisfying the conditions $$\begin{align}\eta_{\mu\nu} \Lambda^\mu_\rho \Lambda^\nu_\sigma = \eta_{\rho\sigma}.\end{align}$$

Lorentz Group

For some purposes, it is useful ot write the Lorentz transformation condition in differenc way. The matrix $\eta_{\mu\nu}$ has an inverse, written $\eta^{\mu\nu}$, which happens to have the same components: it is diagonal, with $\eta^{00}=-1,\ \eta^{11}=\eta^{22}=\eta^{33} = +1$. Multiplying Eq. (5) with $\eta^{\sigma\tau}\Lambda^\kappa_\tau$ and inserting parenthenses judiciously, we have $$\\eta_{\mu\nu} \Lambda^\mu_\rho (\Lambda^\nu_\sigma \Lambda^\kappa_\tau \eta^{\sigma \tau}) = \Lambda^\kappa_\rho = \eta_{\mu\nu} \eta^{\nu\kappa} \Lambda^\mu_\rho.$$

Multiplying with the inverse of the matrix $\eta_{\mu\nu}\Lambda^\mu_\rho$ then gives $$\begin{align}\Lambda^\nu_\sigma \Lambda^\kappa_\tau \eta^{\sigma\tau} = \eta^{\nu\kappa}.\end{align}$$

These transformations form a group. If we first perform a Lorentz transformation (4), and then a second Lorentz transforamtion $x'^\mu\rightarrow x''^\mu$, with $$x''^\mu = \bar{\Lambda}^\mu_\rho x'^\rho + \bar{a}^\mu = \bar{\Lambda}^\rho_\mu(\Lambda^\rho_\nu x^\nu + a^\rho) + \bar{a}^\mu$$ then the effect is the same as the Lorentz transformation $x^\mu\rightarrow x''^\mu$, with $$\begin{align}x''^\mu = (\bar{\Lambda}^\mu_\rho \Lambda^\rho_\nu)x^\nu + (\bar{\Lambda}^\mu_\rho a^\rho + \bar{a}^\nu).\end{align}$$

(Note that if $\Lambda^\mu_\nu$ and $\bar{\Lambda}^\mu_\nu$ both satisfy Eq. (5), then so does $\bar{\Lambda}^\mu_\rho \Lambda^\rho_\nu$, so this is a Lorentz transformation. The bar is used here just to distinguish one Lorentz transformtaion from the other.) The transformations $T(\Lambda,a)$ induced ont physical states therefore satisfy the composition rule $$\begin{align}T(\bar{\Lambda}, \bar{a}) T(\Lambda, a) = T(\bar{\Lambda} \Lambda, \bar{\Lambda} a + \bar{a}).\end{align}$$

Taking the determinant of Eq. (5) gives $$\begin{align}(\det \Lambda )^2 = 1\end{align}$$ so $\Lambda^\mu_\nu$ has an inverse, $(\Lambda^{-1})^\nu_\rho$ which we see from Eq. (5) takes the form $$\begin{align}(\Lambda^{-1})^\rho_\nu = \Lambda_\nu^\rho \equiv \eta_{\nu\mu} \eta^{\rho\sigma}\Lambda^\mu_\sigma.\end{align}$$

The inverse of the transformation $T(\Lambda,a)$ is seen from Eq. (8) to be $T(\Lambda^{-1}, -\Lambda^{-1} a)$, and, of course, the identity transforamtion is $T(1,0)$.

Lorenze transformation on Hilbert Space

In accordanc with the discussion in the previous section, the transformations $T(\Lambda,a)$ induce a unitary linear transformation on vectors in the physical Hilbert space $$\Psi\rightarrow U(\Lambda,a)\Psi.$$

The operators $U$ satisfy a composition rule $$\begin{align}U(\bar{\Lambda}, a) U(\Lambda, a) = U(\bar{\Lambda}\Lambda, \bar{\Lambda} a + \bar{a}).\end{align}$$

(As already mentioned, to avoid the appearance of a phase factor on the right-hand side of Eq. (11), it is , in general, necessary to enlarge the Lorentz group. The appropriate enlargement is described in Section 2.7)

Poincare group

The whole group of transformations $T(\Lambda,a)$ is properly known as the inhomogeneous Lorentz group, or Poincare group. It has a number of important subgroups. First, those transformations with $a^\mu=0$ obiously form a subgroup, with $$\begin{align}T(\bar{\Lambda}, 0) T(\Lambda, 0) = T(\bar{\Lambda} \Lambda, 0)\end{align}$$ known as the homogeneous Lorentz group. Also, we note from Eq. (9) that either $\det \Lambda = +1$ or $\det \Lambda = -1$; those transformations with $\det \Lambda=+1$ obiously form a subgroup of either the homogeneous or the inhomogeneous Lorentz groups. Further, from the $00$-components of Eqs. (5) and (6), we have $$\begin{align} (\Lambda^0_0)^2 = 1 + \Lambda^i_0 \Lambda^i_0 = 1 + \Lambda^0_i \Lambda^0_i.\end{align}$$ with $i$ summed over the values $1,2,$and$3$. We see that either $\Lambda^0_0\ge +1$ or $\Lambda^0_0 \le -1$> Those transformations with $\Lambda^0_0 \ge +1$ form a subgroup. Note that if $\Lambda^\mu_\nu$ and $\bar{\Lambda}^\mu_\nu$ are two such $\Lambda$s, then $$(\bar{\Lambda} \Lambda)^0_0 = \bar{\Lambda}^0_0 \Lambda^0_0 + \bar{\Lambda}^0_1 \Lambda^1_0 + \bar{\Lambda}^0_2 \Lambda^2_0 + \bar{\Lambda}^0_3 \Lambda^3_0.$$

But Eq. (13) shows that the three-vector $(\Lambda^1_0, \Lambda^2_0, \Lambda^3_0)$ has length $\sqrt{(\Lambda^0_0)^2 - 1}$, and similarly the three-vector $(\bar{\Lambda}^0_1, \bar{\Lambda}^2_0, \bar{\Lambda}^0_3)$ has length $\sqrt{(\bar{\Lambda^0_0})^2 - 1}$, so the scalar product of these two three-vectors is bounded by $$\begin{align} \left|\bar{\Lambda}^0_1 \Lambda^1_0 + \bar{\Lambda}^0_2 \Lambda^2_0 + \bar{\Lambda}^0_3 \Lambda^3_0 \right| \le \sqrt{(\Lambda^0_0)^2 - 1} \sqrt{(\bar{\Lambda}^0_0)^2 - 1},\end{align}$$ and so $$(\bar{\Lambda} \Lambda)^0_0 \ge \bar{\Lambda^0_0} \Lambda^0_0 - \sqrt{(\Lambda^0_0)^2 - 1}\sqrt{(\bar{\Lambda}^0_0)^2 - 1} \ge 1.$$

The subgroup of Lorentz transformations with $\det \Lambda=+1$ and $\Lambda^0_0\ge +1$ is known as the proper orthochronous Lorentz group. Since it is not possible by a continuous change of parameters to jump from $\det \Lambda  = +1$ to $\det \Lambda = +1$, or from $\Lambda^0_0 \ge +1$ to $\Lambda^0_0 \le -1$, any Lorentz transformation that can be obtained from the identity by a continuous change of parameters must have $\det \Lambda$ and $\Lambda^0_0$ of the same sign as for the identity, and hence must belong to the proper orthochronous Lorentz group.

Space inversion and Time-reversal

Any Lorentz transformation is either proper and orghochronous, or may be written as the product of an element of the prorper orthochronous Lorentz group with one of the discrete transformations $\mathcal{P}$ or $\mathcal{T}$ or $\mathcal{PT}$, where $\mathcal{P}$ is the space inversion, whose non-zero elements are $$\begin{align}\mathcal{P}^0_0 = 1,\ \mathcal{P}^1_1 = \mathcal{P}^2_2 = \mathcal{P}^3_3 = -1,\end{align}$$ and $\mathcal{T}$ is the time-reversal matrix, whose non-zero elements are $$\begin{align}\mathcal{T}^0_0 = -1,\ \mathcal{T}^1_1 = \mathcal{T}^2_2 = \mathcal{T}^3_3 = 1.\end{align}$$

Thus the study of the whole Lorentz group reduces to the study of its proper orthochronous subgroup, plus space inversion and time-reversal. We will consider space inversion and time-reversal separately in Section 2.6. Until then, we will deal only with the homogeneous or inhomogeneous proper otrhochoronous Lorentz group.