Question

Consider the group of all displacements in three-dimensional space: $$ x^{\prime}=x+a \quad y^{\prime}=y+b \quad z^{\prime}=z+c $$ (a) How many parameters does this group have? (b) Construct the infinitesimal operators (in differential form). (c) Show that all the infinitesimal operators commute with each other.

Short answer

a) 3 parameters. b) Infinitesimal operators are \( L_x = \frac{\text{d}}{\text{d}x} \), \( L_y = \frac{\text{d}}{\text{d}y} \), \( L_z = \frac{\text{d}}{\text{d}z} \). c) They all commute: \( [L_x, L_y] = 0 \), \( [L_y, L_z] = 0 \), \( [L_z, L_x] = 0 \).

Step-by-step solution

Determine the number of parameters

The group of displacements in three-dimensional space is defined by the transformations: \( x' = x + a \), \( y' = y + b \), and \( z' = z + c \). Here, \(a\), \(b\), and \(c\) are constants representing displacements in the \(x\), \(y\), and \(z\) directions, respectively. Each of these constants is an independent parameter, so the group has a total of 3 parameters.

Construct the infinitesimal operators

The infinitesimal displacements can be written as: \( x' = x + \text{d}a \), \( y' = y + \text{d}b \), \( z' = z + \text{d}c \). Considering infinitesimally small displacements, the transformations become: \( x' = x + \text{d}a \approx x + \frac{\text{d}a}{\text{d}t}\text{d}t \), \( y' = y + \text{d}b \approx y + \frac{\text{d}b}{\text{d}t}\text{d}t \), \( z' = z + \text{d}c \approx z + \frac{\text{d}c}{\text{d}t}\text{d}t \). The infinitesimal operators can thus be represented by partial derivatives: \( L_x = \frac{\text{d}}{\text{d}a} \equiv \frac{\text{d}}{\text{d}x} \), \( L_y = \frac{\text{d}}{\text{d}b} \equiv \frac{\text{d}}{\text{d}y} \), \( L_z = \frac{\text{d}}{\text{d}c} \equiv \frac{\text{d}}{\text{d}z} \).

Show that all the infinitesimal operators commute with each other

To show that the infinitesimal operators commute, verify that the commutators are zero. For any two partial derivatives, say \( \frac{\text{d}}{\text{d}x} \) and \( \frac{\text{d}}{\text{d}y} \), the commutator is: \[ [\frac{\text{d}}{\text{d}x}, \frac{\text{d}}{\text{d}y}] \bullet = \frac{\text{d}}{\text{d}x} \frac{\text{d}}{\text{d}y} \bullet - \frac{\text{d}}{\text{d}y} \frac{\text{d}}{\text{d}x} \bullet \] Applying any function \( \bullet \), we get: \( \frac{\text{d}}{\text{d}x} \frac{\text{d}}{\text{d}y} \bullet - \frac{\text{d}}{\text{d}y} \frac{\text{d}}{\text{d}x} \bullet = 0 \), since partial derivatives are commutative. This argument holds similarly for \( \frac{\text{d}}{\text{d}y} \) and \( \frac{\text{d}}{\text{d}z} \), as well as \( \frac{\text{d}}{\text{d}z} \) and \( \frac{\text{d}}{\text{d}x} \). Therefore, \[ [L_x, L_y] = [L_y, L_z] = [L_z, L_x] = 0 \] showing that the infinitesimal operators commute with each other.

Key concepts

Infinitesimal Operators

In physics, infinitesimal operators are crucial tools used to describe small changes or displacements in a system. When we talk about infinitesimal operators in three-dimensional space, we refer to very small shifts or movements.

Consider the transformation equations: \( x' = x + da \), \( y' = y + db \), and \( z' = z + dc \), where \(da\), \(db\), and \(dc\) represent infinitesimal displacements.

To derive actions for these small changes, we use derivatives. The infinitesimal operators become:

• \( L_x = \frac{d}{dx} \)
• \( L_y = \frac{d}{dy} \)
• \( L_z = \frac{d}{dz} \)

These operators help us describe and calculate the impact of tiny shifts in the x, y, and z directions.

Commutators

Commutators are operators used to measure the non-commutativity of two operators. Non-commutativity means that the order in which you apply two operations matters.

For infinitesimal operators, the commutator is defined by:
\[ [L_x, L_y] = L_x L_y - L_y L_x \]
where \( L_x \) and \( L_y \) are the infinitesimal operators for different directions.

In this context, infinitesimal operators commute if their commutator is zero. For example:
\[ [ \frac{d}{dx}, \frac{d}{dy} ] \bullet = \frac{d}{dx} \frac{d}{dy} \bullet - \frac{d}{dy} \frac{d}{dx} \bullet = 0 \]
Applying any function \( \bullet \), the result is zero, indicating that the two operations are commutative. Similarly, this holds for all pairs of partial derivative operators.

Three-Dimensional Displacements

Three-dimensional displacements involve shifts in the x, y, and z directions. These are fundamental in physics for describing the motion of objects or transformations in space.

The transformations can be written as:

• \( x' = x + a \)
• \( y' = y + b \)
• \( z' = z + c \)

Here, \(a\), \(b\), and \(c\) are constants representing the displacements. These displacements are captured by three parameters, making the system three-dimensional.

Infinitesimal displacements can then be viewed as:
\( x' = x + da \), \( y' = y + db \), and \( z' = z + dc \), where \(da\), \(db\), and \(dc\) are very small changes.

Partial Derivatives

Partial derivatives are used to describe how a function changes as one of its variables changes, keeping others constant. They are essential in multivariable calculus and physics.

The partial derivative of a function \( f(x, y, z) \) with respect to x is denoted as \( \frac{\partial f}{\partial x} \), evaluating how \( f \) changes as \( x \) changes, while y and z remain fixed.

• \( L_x = \frac{\partial }{\partial x} \)
• \( L_y = \frac{\partial }{\partial y} \)
• \( L_z = \frac{\partial }{\partial z} \)

These partial derivatives or infinitesimal operators help in analyzing small displacements and understanding how different variables interact in a function. They are also key in proving that certain operators commute, as shown above by verifying commutators.