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# 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Discrete transformations

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## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Discrete transformations

Lorentz transformations are defined to be those that preserve the Minkowski metric:
$$\Lambda^T g \Lambda=g$$
Equivalently, they are those that leave inner products such as
$$V_\mu W^\mu=V_0 W_0-V_1 W_1-V_2 W_2-V_3 W_3$$
invariant. By this definition, the transformations
$$P:(t, x, y, z) \rightarrow(t,-x,-y,-z)$$
known as parity and
$$T:(t, x, y, z) \rightarrow(-t, x, y, z)$$
known as time reversal are also Lorentz transformations. They can be written as
$$P=\left(\begin{array}{cccc} 1 & & & \ & -1 & & \ & & -1 & \ & & & -1 \end{array}\right), \quad T=\left(\begin{array}{cccc} -1 & & & \ & 1 & & \ & & 1 & \ & & & 1 \end{array}\right)$$
Parity and time reversal are special because they cannot be written as the product of rotations and boosts, Eqs. (2.13) and (2.14). Discrete transformations play an important role in quantum field theory (see Chapter 11).
We say that a vector is timelike when
$$V^\mu V_\mu>0 \quad \text { (timelike) }$$
and spacelike when
$$V^\mu V_\mu<0 \quad \text { (spacelike) }$$
Naturally, time $=(t, 0,0,0)$ is timelike and space $=(0, x, 0,0)$ is spacelike. Whether something is timelike or spacelike is preserved under Lorentz transformations since the norm is preserved. If a vector has zero norm we say it is lightlike:
$$V^\mu V_\mu=0 \quad \text { (lightlike). }$$

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Solving problems with Lorentz invariance

Special relativity in quantum field theory is much easier than the special relativity you learned in your introductory physics course. We never need to talk about putting long cars in small garages or engineers with flashlights on trains. These situations are all designed to make your non-relativistic intuition mislead you. In quantum field theory, other than the perhaps unintuitive notion that energy can turn into matter through $E=m c^2$, your non-relativistic intuition will serve you perfectly well.

For field theory, all you really need from special relativity is the one equation that defines Lorentz transformations:
$$\Lambda^T g \Lambda=g$$
This implies that contractions such as $p^2 \equiv p^\mu p_\mu$ are Lorentz invariant. For problems that involve changing frames, usually you know everything in one frame and are interested in some quantity in another frame. For example, you may know momenta $p_1^\mu$ and $p_2^\mu$ of two incoming particles that collide and are interested in the energy of an outgoing particle $E_3$ in the center-of-mass frame (the center-of-mass frame is defined as the frame in which the total 3-momenta, $\vec{p}{\text {tot }}=0$ ). For such problems, it is best to first calculate a Lorentzinvariant quantity such as $p{\text {tot }}^2=\left(p_1^\mu+p_2^\mu\right)^2$ in the first frame, then go to the second frame, and solve for the unknown quantity. Since $p_{\text {tot }}^2$ is Lorentz invariant, it has the same value in both frames. Usually, when you input everything you know about the second frame (e.g. $\vec{p}{\text {tot }}=0$ if it is the center-of-mass frame), you can solve for the remaining unknowns. If you find yourself plugging in explicit boost and rotation matrices, you are probably solving the problem the hard way. This trick is especially useful for situations in which there are many particles, say $p_1^\mu, \ldots, p_5^\mu$, and therefore many Lorentz-invariant quantities, such as $p_1^\mu p{4 \mu}$ or $\left(p_5^\mu+p_4^\mu\right)^2$

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Discrete transformations

$$\Lambda^T g \Lambda=g$$

$$V_\mu W^\mu=V_0 W_0-V_1 W_1-V_2 W_2-V_3 W_3$$

$$P:(t, x, y, z) \rightarrow(t,-x,-y,-z)$$

$$T:(t, x, y, z) \rightarrow(-t, x, y, z)$$

$$P=\left(\begin{array}{cccc} 1 & & & \ & -1 & & \ & & -1 & \ & & & -1 \end{array}\right), \quad T=\left(\begin{array}{cccc} -1 & & & \ & 1 & & \ & & 1 & \ & & & 1 \end{array}\right)$$

$$V^\mu V_\mu>0 \quad \text { (timelike) }$$

$$V^\mu V_\mu<0 \quad \text { (spacelike) }$$

$$V^\mu V_\mu=0 \quad \text { (lightlike). }$$

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Solving problems with Lorentz invariance

$$\Lambda^T g \Lambda=g$$

## MATLAB代写

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