Bogoliubov Transformations for Creation/Annihilation Operators

1. The General Transformation of Modes and Operators

Let's formalize the setup from our last discussion. We have a scalar field $\varphi$ that satisfies the Klein-Gordon equation in a curved spacetime.

• An observer, "Alice," chooses a complete orthonormal basis of solutions $\{u_i\}$ to define her notion of particles. She decomposes the field operator $\hat{\varphi }$ as:
  $\hat{\varphi }(x)={\sum }_i\left({\hat{a}}_i{u}_i(x)+{\hat{a}}_i^{†}{u}_i^{\ast }(x)\right)$
  Her vacuum, $|0_a⟩$, is defined by ${\hat{a}}_i|0_a⟩=0$ for all $i$.

• Another observer, "Bob," chooses a different complete orthonormal basis $\{{\bar{u}}_j\}$ and decomposes the same field operator as:
  $\hat{\varphi }(x)={\sum }_j\left({\hat{b}}_j{\bar{u}}_j(x)+{\hat{b}}_j^{†}{\bar{u}}_j^{\ast }(x)\right)$
  His vacuum, $|0_b⟩$, is defined by ${\hat{b}}_j|0_b⟩=0$ for all $j$.

Since both $\{u_i\}$ and $\{{\bar{u}}_j\}$ are complete bases for the space of solutions, any mode in one basis can be expressed as a linear combination of the modes in the other. This is the heart of the Bogoliubov transformation.

Let's summarize the key relationships you've just studied. A mode ${\bar{u}}_j$ from Bob's basis can be written in terms of Alice's basis functions as:
${\bar{u}}_j={\sum }_i\left({\alpha }_{ji}{u}_i+{\beta }_{ji}{u}_i^{\ast }\right)$
The coefficients are given by the Klein-Gordon inner product, which we introduced previously:
${\alpha }_{ji}=⟨u_i,{\bar{u}}_j⟩\text{and}{\beta }_{ji}=-⟨u_i^{\ast },{\bar{u}}_j⟩$
The $\alpha$ coefficients measure the overlap between the two sets of positive-frequency modes, while the crucial $\beta$ coefficients measure the overlap between one observer's positive-frequency modes and the other's negative-frequency modes.

This leads directly to the transformation between the operators. Equating the two expansions for $\hat{\varphi }$ and solving for the operators gives (as shown in Eq. 3.76 of the notes):
${\hat{b}}_j={\sum }_i\left({\alpha }_{ji}^{\ast }{\hat{a}}_i-{\beta }_{ji}^{\ast }{\hat{a}}_i^{†}\right)$
${\hat{b}}_j^{†}={\sum }_i\left({\alpha }_{ji}{\hat{a}}_i^{†}-{\beta }_{ji}{\hat{a}}_i\right)$
This is the Bogoliubov transformation for the creation and annihilation operators. It explicitly shows that Bob's annihilation operator ${\hat{b}}_j$ is a mixture of Alice's annihilation operator ${\hat{a}}_i$ and, critically, her creation operator ${\hat{a}}_i^{†}$.

2. Preserving the Structure of Quantum Mechanics

The operators for both Alice and Bob must describe the same physics. Therefore, Bob's operators ${\hat{b}}_j$ and ${\hat{b}}_j^{†}$ must satisfy the same canonical commutation relations as Alice's operators:
$[{\hat{b}}_i,{\hat{b}}_j]=0,[{\hat{b}}_i^{†},{\hat{b}}_j^{†}]=0,[{\hat{b}}_i,{\hat{b}}_j^{†}]={\delta }_{ij}$
Substituting the transformation laws into these relations imposes strong constraints on the Bogoliubov coefficients $\alpha$ and $\beta$. For instance, demanding that $[{\hat{b}}_i,{\hat{b}}_j^{†}]={\delta }_{ij}$ leads to the condition:
${\sum }_k\left({\alpha }_{ik}^{\ast }{\alpha }_{jk}-{\beta }_{ik}^{\ast }{\beta }_{jk}\right)={\delta }_{ij}$
This is a normalization condition. In a more compact matrix notation, where $\alpha$ and $\beta$ are treated as matrices, this becomes:
$\alpha {\alpha }^{†}-\beta {\beta }^{†}=\mathbf{I}$
This relation is a fundamental property of Bogoliubov transformations, ensuring that the quantum structure of the theory (the commutation relations) is preserved when changing basis. You can find these relations as equations (3.77) and (3.82) in the resource you just read.

3. A Toy Model: The Parametrically Excited Oscillator

The full field theory can seem abstract. To build intuition, we can study a much simpler system that captures the essential physics: a single quantum harmonic oscillator whose frequency $\omega (t)$ changes with time. This is a powerful analogy for a quantum field mode propagating through an expanding universe, where the expansion of space (the scale factor $a(t)$) effectively changes the mode's frequency.

Imagine an oscillator that starts in the distant past ($t\to -\infty$) with a constant frequency ${\omega }_{in}$ and ends in the distant future ($t\to +\infty$) with a constant frequency ${\omega }_{out}$.

• In the "in" region, we can define a vacuum state $|0_{in}⟩$ based on the ground state of the oscillator with frequency ${\omega }_{in}$.
• In the "out" region, we define an "out" vacuum $|0_{out}⟩$ based on the ground state for frequency ${\omega }_{out}$.

If we start the system in $|0_{in}⟩$, what state will it be in at late times, as described by an observer in the "out" region? This is precisely the question the Bogoliubov transformation answers.

As the notes show, the number of "out" quanta an observer would measure if the system was initially in the "in" vacuum is:
$⟨0_{in}|N_{out}|0_{in}⟩=⟨0_{in}|a_{out}^{†}{a}_{out}|0_{in}⟩=|\beta {|}^2$
This confirms our general result in a simple, concrete model. If the frequency changes with time (${\omega }_{in}\ne {\omega }_{out}$), then $\beta \ne 0$, and the initial vacuum state becomes populated with particles. This phenomenon is called parametric excitation.

4. The Structure of the Vacuum: Squeezed States

We can go further than just calculating the average number of particles. What is the full quantum state of $|0_{in}⟩$ when expressed in the basis of "out" energy eigenstates? The Bogoliubov transformation allows us to find the exact answer. The result is that the "in" vacuum is what is known in quantum optics as a squeezed vacuum state from the "out" observer's perspective.

The derivation shows that the "in" vacuum is a superposition of "out" states containing only pairs of particles:
$|0_{in}⟩=N{\sum }_n\frac{\sqrt{(2n)!}}{n!}{\left(-\frac{\bar{\beta }}{2\alpha }\right)}^n|2n{⟩}_{out}$
This is the characteristic form of a squeezed state. For a quantum field, this means particles are always created in pairs with opposite momenta ($k,-k$) to conserve momentum, as shown in resource LINK, Eq. (5.42). The fact that the relationship between the two vacua can be described by a unitary operator, the squeeze operator $S$, underscores that this is a fully consistent quantum mechanical transformation.

Conclusion

In this lesson, we have formalized the connection between different observers' descriptions of a quantum field. This is a cornerstone of QFT in curved spacetime.

Key Takeaways:

• Different choices of basis modes $\{u_i\}$ and $\{{\bar{u}}_j\}$ lead to different sets of creation/annihilation operators.
• The relationship between these operators is a Bogoliubov transformation, parameterized by coefficients $\alpha$ and $\beta$.
• The $\beta$ coefficient quantifies the mixing of positive and negative frequencies between the two bases. A non-zero $\beta$ is the signal of particle ambiguity and creation.
• These transformations must satisfy constraints, such as $\alpha {\alpha }^{†}-\beta {\beta }^{†}=\mathbf{I}$, to preserve the canonical commutation relations.
• One observer's vacuum state appears as a squeezed state to another, containing particles created in pairs.

Preview of the Next Lesson:

We have now developed the essential mathematical machinery. In the next lesson, we will apply it to our first major physical prediction: the Unruh effect. We will perform a concrete calculation of the Bogoliubov coefficients that relate an inertial observer in Minkowski space to a uniformly accelerating one. This will allow us to derive the remarkable result that the accelerating observer perceives the empty Minkowski vacuum as a thermal bath of particles at a specific temperature proportional to their acceleration.