Direct HepLorentzVector Boosts and Rotations

Direct boosts and rotations are methods of HepLorentzVector which modify the 4-vector being acted upon-- e.g., w.boost $(\hat{u}, \beta)$--or global functions which form a new vector-- e.g., boostOf $(w, \hat{u}, \beta)$.

Rotations act in the obvious manner, affecting only the $\vec{v}$ component of the 4-vector--see §.

In analogy with our ``active'' rotation viewpoint, boosts are treated as ``active'' transformations rather than transformations of the coordinate system. That is, if you take a 4-vector at rest, with positive mass ($t$), and boost it by a positive amount in the X direction, the resulting 4-vector will have positive $x$.

Boosts along the X, Y, or Z axis are simpler than the general case. Let $w = (\vec{v}, t) = (x, y, z, t)$:

$$w.\mbox{boostX} (\beta) \Longrightarrow ( \gamma x + \beta \gamma t, y, z, \gamma t + \beta \gamma x )$$ (100)

$$w.\mbox{boostY} (\beta) \Longrightarrow ( x, \gamma y + \beta \gamma t, z, \gamma t + \beta \gamma y )$$ (101)

$$w.\mbox{boostZ} (\beta) \Longrightarrow ( x, y, \gamma z + \beta \gamma t, \gamma t + \beta \gamma z )$$ (102)

$$\gamma \equiv \frac{1}{\sqrt{1-\beta^2}}$$

More general rotations boosts may be expressed in terms of $\beta$ along an axis given as a Hep3Vector $\hat{u}$ (which will be normalized), or in terms of a Hep3Vector boost $\vec{\beta}$, which must obey $\vert\vec{\beta}\vert<1$. (Boosts beyond the speed of light ZMthrow a ZMxpvTachyonic error, and leave the 4-vector unchanged if this is ignored.)

For the axis $(\hat{u}, \beta)$ form,

$$\left\{ \begin{array}{lcl} t & \longleftarrow & \gamma t + \beta ... ...\beta \vec{v} \cdot \hat{u} + \beta \gamma t \right] \hat{u} \end{array}\right.$$ (103)

$$\gamma \equiv \frac{1}{\sqrt{1-\beta^2}}$$

For the boost vector $(\vec{\beta})$ form,

$$\left\{ \begin{array}{lcl} t & \longleftarrow & \gamma t + \gamma... ...^2} \vec{v} \cdot \vec{\beta} + \gamma t \right] \vec{\beta} \end{array}\right.$$ (104)

$$\gamma \equiv \frac{1}{\sqrt{1-\vert\vec{\beta}\vert^2}}$$