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Ch. 5 Solutions

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Basic

Medium

Basic

B-1. Metric in Light-Cone Coordinates

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Problem:

From the definition of light-cone coordinates \(x^\pm = (x^0 \pm x^1)/\sqrt{2}\), show by direct calculation that

\[ ds^2 = -(dx^0)^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2 \]

can be rewritten as

\[ ds^2 = -2\,dx^+ dx^- + (dx^2)^2 + (dx^3)^2 \]

(with \(c = 1\)).

Solution strategy: Calculate \(dx^+ dx^-\) from the definition of light-cone coordinates and obtain the relationship with \((dx^0)^2 - (dx^1)^2\).

Calculation:

From the definition of light-cone coordinates

\[ dx^+ = \frac{1}{\sqrt{2}}(dx^0 + dx^1), \qquad dx^- = \frac{1}{\sqrt{2}}(dx^0 - dx^1) \]

Computing the product

\[ dx^+ dx^- = \frac{1}{2}(dx^0 + dx^1)(dx^0 - dx^1) = \frac{1}{2}\left[(dx^0)^2 - (dx^1)^2\right] \]

That is

\[ (dx^0)^2 - (dx^1)^2 = 2\,dx^+ dx^- \]

Substituting this into \(ds^2 = -(dx^0)^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2\):

\[ ds^2 = -[(dx^0)^2 - (dx^1)^2] + (dx^2)^2 + (dx^3)^2 = -2\,dx^+ dx^- + (dx^2)^2 + (dx^3)^2 \]
\[ \boxed{ds^2 = -2\,dx^+ dx^- + (dx^2)^2 + (dx^3)^2} \]

Verification: If \(dx^2 = dx^3 = 0\) and \(dx^0 = dx^1\) (light traveling in the positive \(x^1\) direction), then \(dx^- = 0\) so \(ds^2 = 0\). ✓

B-2. Matrix Representation of the Light-Cone Metric

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Problem:

Find the inverse matrix \(\hat{\eta}^{\mu\nu}\) of the light-cone metric matrix (with row and column ordering \(+, -, 2, 3\))

\[ \hat{\eta}_{\mu\nu} = \begin{pmatrix} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

and verify that it satisfies \(\hat{\eta}^{\mu\lambda}\hat{\eta}_{\lambda\nu} = \delta^\mu{}_\nu\).

Solution strategy: The \((+,-)\) block is a \(2 \times 2\) anti-diagonal matrix. Computing the inverse from the definition of the cofactor matrix, we find it coincides with the original matrix. The \((2,3)\) block is the identity matrix and remains unchanged.

Calculation:

Let the \(2 \times 2\) block for \((+, -)\) be

\[ M = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \]

Since the inverse of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is \(\frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\),

\[ M^{-1} = \frac{1}{(0)(0) - (-1)(-1)}\begin{pmatrix} 0 & -(-1) \\ -(-1) & 0 \end{pmatrix} = \frac{1}{-1}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = M \]

That is, \(M\) is an involutory matrix (equivalent to \(M^2 = I\)). The \((2, 3)\) block is the identity matrix and remains unchanged. Therefore

\[ \boxed{\hat{\eta}^{\mu\nu} = \begin{pmatrix} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \hat{\eta}_{\mu\nu}} \]

Written in components: \(\hat{\eta}^{+-} = \hat{\eta}^{-+} = -1\), \(\hat{\eta}^{22} = \hat{\eta}^{33} = +1\), and all others are zero.

Verification of \(\hat{\eta}^{\mu\lambda}\hat{\eta}_{\lambda\nu} = \delta^\mu{}_\nu\): For example, \((\mu, \nu) = (+, +)\):

\[ \hat{\eta}^{+\lambda}\hat{\eta}_{\lambda+} = \hat{\eta}^{+-}\hat{\eta}_{-+} = (-1)(-1) = 1 = \delta^+{}_+ \quad \checkmark \]

\((\mu, \nu) = (+, -)\):

\[ \hat{\eta}^{+\lambda}\hat{\eta}_{\lambda-} = \hat{\eta}^{+-}\hat{\eta}_{--} = (-1)(0) = 0 = \delta^+{}_- \quad \checkmark \]

The other components can be verified similarly.

Note: In light-cone coordinates, raising and lowering indices does not correspond to a time/space sign flip as in the usual coordinates, but rather to a swap of \(+\) and \(-\): \(A_+ = -A^-\), \(A_- = -A^+\), \(A_i = A^i\) (\(i = 2, 3\)).

B-3. Four-Momentum in Light-Cone Coordinates and \(p^-\)

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Problem:

Express the four-momentum \(p^\mu = (p^0, p^1, p^2, p^3)\) of a particle with mass \(m\) in terms of light-cone coordinate components \(p^\pm = (p^0 \pm p^1)/\sqrt{2}\), \(p^2, p^3\).

(a) Write the invariant norm \(p^\mu p_\mu\) of the four-momentum in light-cone coordinates. Show that the result is \(-2 p^+ p^- + (p^2)^2 + (p^3)^2\).

(b) Using \(p^\mu p_\mu = -m^2\), solve for \(p^-\) in terms of \(p^+, p^2, p^3, m\) and derive

\[ p^- = \frac{(p^2)^2 + (p^3)^2 + m^2}{2\,p^+} \]

(c) In ordinary coordinates, solving \(p^\mu p_\mu = -m^2\) for \(p^0\) gives \(p^0 = \pm\sqrt{|\vec{p}|^2 + m^2}\), leaving a \(\pm\) sign ambiguity. In light-cone coordinates, \(p^-\) is uniquely determined as shown in (b). Explain this difference physically.

(a) Light-Cone Coordinate Expression of the Invariant Norm:

Calculation:

In ordinary coordinates,

\[ p^\mu p_\mu = \eta_{\mu\nu}p^\mu p^\nu = -(p^0)^2 + (p^1)^2 + (p^2)^2 + (p^3)^2 \]

By the same calculation as Problem 5.1, \((p^0)^2 - (p^1)^2 = 2 p^+ p^-\), so

\[ \boxed{p^\mu p_\mu = -2\,p^+ p^- + (p^2)^2 + (p^3)^2} \]

(b) Derivation of \(p^-\):

Substituting \(p^\mu p_\mu = -m^2\) into (a):

\[ -2 p^+ p^- + (p^2)^2 + (p^3)^2 = -m^2 \]

Solving for \(p^-\):

\[ 2 p^+ p^- = (p^2)^2 + (p^3)^2 + m^2 \]
\[ \boxed{p^- = \frac{(p^2)^2 + (p^3)^2 + m^2}{2\,p^+}} \]

(c) Physical Interpretation of the Disappearance of Sign Ambiguity:

Ordinary coordinates: Solving \(p^\mu p_\mu = -m^2\) for \(p^0\) gives a quadratic equation \((p^0)^2 = |\vec{p}|^2 + m^2\), yielding \(p^0 = \pm\sqrt{|\vec{p}|^2 + m^2}\). Both positive-energy and negative-energy solutions emerge. In QFT, the negative-energy solutions are physically interpreted as antiparticles, but the quantization procedure requires separately specifying how to handle each sign of the solution.

Light-cone coordinates: On the other hand, in (a), \(p^\mu p_\mu = -2 p^+ p^- + (p^2)^2 + (p^3)^2\) contains \(p^-\) only to first order. Therefore, \(p^\mu p_\mu = -m^2\) is a first-order equation in \(p^-\), and as shown in (b), \(p^-\) is uniquely determined from \(p^+, p^2, p^3, m\). There is no sign ambiguity. The physically meaningful "forward light cone \(p^0 > 0\)" corresponds to the region \(p^+ > 0\), \(p^- > 0\) (when \(m^2 \geq 0\), choosing \(p^+ > 0\) automatically gives \(p^- > 0\) from (b)), so once the propagation direction \(p^+ > 0\) of the particle is specified, only positive-energy states are automatically selected.

This is the core reason why the treatment of antiparticles and the handling of negative-norm states become simpler in light-cone quantization of strings (Ch. 14). The trade-off is that Lorentz covariance is no longer manifest (since \(p^+\) is treated specially), but physical results remain invariant.

Consistency check: For \(m = 0\) (photon), \(p^- = [(p^2)^2 + (p^3)^2]/(2 p^+)\). In particular, for a photon with zero transverse momentum (\(p^2 = p^3 = 0\)), we get \(p^- = 0\), i.e., \(p^0 = p^1\), which is consistent with light propagating in the positive \(x^1\) direction at the speed of light \(c = 1\). ✓

Medium

M-1. Inner Product in Light-Cone Coordinates

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Problem:

Express the Minkowski inner product \(A^\mu B_\mu = \eta_{\mu\nu}A^\mu B^\nu\) of two 4-vectors \(A^\mu, B^\mu\) in terms of the light-cone coordinate components \(A^\pm, A^2, A^3\) and \(B^\pm, B^2, B^3\).

Solution strategy: Rewrite \(A^\mu B_\mu = -A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3\). Inverting the definitions \(A^\pm = (A^0 \pm A^1)/\sqrt{2}\) gives \(A^0 = (A^+ + A^-)/\sqrt{2}\), \(A^1 = (A^+ - A^-)/\sqrt{2}\). Substitute these (and similarly for \(B\)).

Calculation:

Expanding \(A^0 B^0\) in light-cone components:

\[ A^0 B^0 = \frac{1}{2}(A^+ + A^-)(B^+ + B^-) = \frac{1}{2}(A^+ B^+ + A^+ B^- + A^- B^+ + A^- B^-) \]

Similarly for \(A^1 B^1\):

\[ A^1 B^1 = \frac{1}{2}(A^+ - A^-)(B^+ - B^-) = \frac{1}{2}(A^+ B^+ - A^+ B^- - A^- B^+ + A^- B^-) \]

Taking the difference, the \(A^+ B^+\) and \(A^- B^-\) terms cancel, leaving only the cross terms:

\[ -A^0 B^0 + A^1 B^1 = -(A^+ B^- + A^- B^+) \]

Therefore

\[ \boxed{A^\mu B_\mu = -(A^+ B^- + A^- B^+) + A^2 B^2 + A^3 B^3} \]

Alternative expression: Expanding \(A^\mu B_\mu = \hat{\eta}_{\mu\nu}A^\mu B^\nu\) using the light-cone metric \(\hat{\eta}_{\mu\nu}\), the non-zero components are \(\hat{\eta}_{+-} = \hat{\eta}_{-+} = -1\), \(\hat{\eta}_{22} = \hat{\eta}_{33} = 1\), so

\[ A^\mu B_\mu = \hat{\eta}_{+-}A^+ B^- + \hat{\eta}_{-+}A^- B^+ + A^2 B^2 + A^3 B^3 = -A^+ B^- - A^- B^+ + A^2 B^2 + A^3 B^3 \]

This gives the same result.

Consistency check: Setting \(A = B\) gives \(A^\mu A_\mu = -2 A^+ A^- + (A^2)^2 + (A^3)^2\), which agrees with Problem 5.1. ✓

M-2. Lorentz Transformation (Boost) in Light-Cone Coordinates

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Problem:

A boost in the \(x^1\) direction (where the \(S'\) frame moves with velocity \(v\) in the \(x^1\) direction relative to the \(S\) frame) can be written using the rapidity \(\varphi\) (\(\tanh\varphi = v\)) as

\[ \begin{pmatrix} x^{0'} \\ x^{1'} \end{pmatrix} = \begin{pmatrix} \cosh\varphi & -\sinh\varphi \\ -\sinh\varphi & \cosh\varphi \end{pmatrix} \begin{pmatrix} x^0 \\ x^1 \end{pmatrix} \]

(see General Relativity 3.3 "Derivation of the Lorentz Transformation"). Rewrite this in light-cone coordinates \(x^\pm\) and show the following.

(a) \(x^{+\prime} = e^{-\varphi}\,x^+\), \(x^{-\prime} = e^{\varphi}\,x^-\) (a boost in the \(x^+\) direction acts as a scale transformation in light-cone coordinates).

(b) Therefore, the product \(x^+ x^-\) is invariant. Verify that this is consistent with the invariance of the \(-2\,dx^+ dx^-\) part of \(ds^2\).

(a) Proof that it becomes a scale transformation:

Calculation:

A boost in the \(x^1\) direction with rapidity \(\varphi\) is

\[ x^{0'} = \cosh\varphi\cdot x^0 - \sinh\varphi\cdot x^1 \]
\[ x^{1'} = -\sinh\varphi\cdot x^0 + \cosh\varphi\cdot x^1 \]

Converting to light-cone coordinates:

\[ x^{+\prime} = \frac{1}{\sqrt{2}}(x^{0'} + x^{1'}) = \frac{1}{\sqrt{2}}\left[(\cosh\varphi - \sinh\varphi)x^0 + (\cosh\varphi - \sinh\varphi)x^1\right] \]
\[ = (\cosh\varphi - \sinh\varphi)\cdot\frac{x^0 + x^1}{\sqrt{2}} = e^{-\varphi}\,x^+ \]

where we used \(\cosh\varphi - \sinh\varphi = e^{-\varphi}\).

Similarly

\[ x^{-\prime} = \frac{1}{\sqrt{2}}(x^{0'} - x^{1'}) = \frac{1}{\sqrt{2}}\left[(\cosh\varphi + \sinh\varphi)x^0 - (\cosh\varphi + \sinh\varphi)x^1\right] \]
\[ = (\cosh\varphi + \sinh\varphi)\cdot\frac{x^0 - x^1}{\sqrt{2}} = e^{\varphi}\,x^- \]
\[ \boxed{x^{+\prime} = e^{-\varphi}\,x^+, \qquad x^{-\prime} = e^{\varphi}\,x^-} \]

(b) Invariance of \(x^+ x^-\):

From (a):

\[ x^{+\prime}\,x^{-\prime} = (e^{-\varphi}\,x^+)(e^{\varphi}\,x^-) = e^{-\varphi + \varphi}\,x^+ x^- = x^+ x^- \]

Therefore \(x^+ x^-\) is invariant under boosts in the \(x^1\) direction.

Consistency with \(ds^2\): The same holds in differential form: \(dx^{+\prime}\,dx^{-\prime} = dx^+ dx^-\). Also, \(x^2, x^3\) are unchanged under an \(x^1\) boost, so \((dx^{2\prime})^2 + (dx^{3\prime})^2 = (dx^2)^2 + (dx^3)^2\). Therefore

\[ ds^{\prime 2} = -2\,dx^{+\prime}\,dx^{-\prime} + (dx^{2\prime})^2 + (dx^{3\prime})^2 = -2\,dx^+ dx^- + (dx^2)^2 + (dx^3)^2 = ds^2 \]

This is consistent with the invariance of the spacetime interval. ✓

Geometric interpretation: In the usual \((x^0, x^1)\) plane, a Lorentz boost appears as a hyperbolic rotation (involving \(\cosh, \sinh\)), whereas in the \((x^+, x^-)\) plane, a boost becomes a scale transformation along the axes. This is more straightforward than the hyperbolic formulation, and it is immediately apparent that the light-cone directions are preserved. This is another advantage of light-cone coordinates.