Problem 5 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

Show the 'anticonmutation law' of conjugation, $$ \overline{P Q}=\bar{Q} \bar{P} $$ Hence prove $$ |P Q|=|P||Q| . $$

Step-by-Step Solution

Answer

The anticommutation law \( \overline{P Q}=\overline{Q}\overline{P} \) and modulus property \( |P Q|=|P||Q| \) of complex numbers are both true.

Step 1: Prove the Anticommutation law

To prove this, let's say \( P = a + ib \) and \( Q = c + id \). Now, \( PQ = (ac − bd) + i(ad + bc) \). The conjugate of \( PQ \) is \( \overline{P Q} = (ac - bd) - i(ad + bc) \). On the other hand, \( \overline{Q} \overline{P} = (a - ib)(c - id) = (ac - bd) - i(ad + bc) \) which is equal to \( \overline{P Q} \). Hence the anticommutation law is proven.

Step 2: Prove the Modulus Properties

Now to prove \( |P Q| = |P||Q| \), let \( P = a + ib \) and \( Q = c + id \). Then \( |P|^2 = a^2 + b^2 \), \( |Q|^2 = c^2 + d^2 \) and \( (PQ) = (ac - bd)^2 + (ad + bc)^2 \) which simplifies to \( a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 = (a^2 + b^2)(c^2 + d^2) = |P|^2|Q|^2 \). Taking square root on both sides we finally get \( |P Q| = |P||Q| \). Hence the modulus property is proven.

Key Concepts

Anticommutation Law

The anticommutation law is a fascinating concept in the realm of complex numbers. It states how the conjugate of the product of two complex numbers is equal to the product of their individual conjugates in reverse order. When we express this mathematically, it looks like this:

• \( \overline{P Q} = \overline{Q} \overline{P} \)

This means if you take two complex numbers, say \( P = a + ib \) and \( Q = c + id \), multiply them, and find the conjugate of this product, it is the same as finding the conjugates separately and then multiplying them in reverse. Let’s break it down further:

• First, compute \( PQ = (ac − bd) + i(ad + bc) \)
• Next, the conjugate of \( PQ \) becomes \( (ac - bd) - i(ad + bc) \)
• Finally, observe that \( \overline{Q} \overline{P} = (a - ib)(c - id) = (ac - bd) - i(ad + bc) \)

By equating these expressions, we affirm the anticommutation law. This property simplifies many complex number manipulations by reducing the calculations involved.

Complex Conjugation

Complex conjugation is a fundamental operation in complex numbers. The conjugate of a complex number is achieved by changing the sign of the imaginary part. For example, if you have \( a + ib \), its complex conjugate is \( a - ib \). The beauty of this operation is that it helps in finding the inverse or solving equations involving complex numbers.

• Conjugation reflects the number across the real axis in the complex plane.
• It is used to simplify division of complex numbers.

When working with products like \( \overline{P Q} \), remembering the step to reverse the order of conjugation (as in anticommutation) is key. Conjugation is not just a theoretical exercise; it's utilized in practical scenarios too, such as in physics and engineering, to simplify problems and provide real-world solutions.

Modulus of a Complex Number

The modulus of a complex number is a measure of its size, or absolute value, in the complex plane. It gives the distance from the point to the origin and is calculated using the formula:

• For a complex number \( P = a + ib \), the modulus is \( |P| = \sqrt{a^2 + b^2} \)

The modulus is a non-negative real number, even though it applies to complex numbers. When dealing with the product of complex numbers, the modulus changes as follows:

• The modulus of the product \( |PQ| \) meets the property \( |P Q| = |P||Q| \)
• This simply means: multiplying the moduli of two numbers gives the modulus of the product.

This property is very useful because it aligns with our intuition from real number arithmetic and allows us to simplify problems involving complex number multiplication.