Canonical commutation relation, from QM to QFT.

• annihilatorM
In summary: Putting it all together, you get$$[\dot q(x,t),q(x',t)] = -\frac{i}{\mu}\int_{-\frac12\Delta x}^{\frac12\Delta x} dy\delta(y+x-x') + O(\Delta x^2).$$In summary, the quantization condition for a system of $n$ coupled harmonic oscillators in 1 dimension can be expressed using the new variable $q(x,t)$ instead of $q_j(t)$. The commutation relation between $\dot q(x,t)$ and $q(x',t)$ can be obtained by considering the two cases where $x=x'$ and $|x-x'|=j\Delta x$, leading to an integral
annihilatorM

Homework Statement

This is a system of n coupled harmonic oscillators in 1 dimension.
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Since the distance between neighboring oscillators is ## \Delta x ## one can characterize the oscillators equally well by ## q(x,t) ## instead of ## q_j(t) ##. Then ## q_{j \pm 1} ## should be replaced by ## q(x \pm \Delta x , t) ##. Show that the quantization condition

$$[\dot{q}_j(t), q_k (t)] = - \frac{i}{\mu} \delta _{jk}$$

can be expressed as$$[ \dot{q} (x,t), q (x',t)] = - \frac{i}{\mu} \int_{ \frac{- \Delta x}{2}}^{\frac{ \Delta x}{2}} dy \delta (y-x'+x)$$

Hint: Distinguish the two cases ## x=x' ## and ## | x-x' | = j \Delta x ##

The Attempt at a Solution

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Making the substitution of ## q(x,t) ## in place of ## q_j (t) ## I obtain

$$[ \dot{q} (x,t), q (x',t)] = \delta (x-x')$$

which is the expected commutation relation I've seen from other sources, however, I don't understand why the integration with respect to y is introduced or where the integration bounds come from. What is wrong with simply writing down what I have above?

A:You have an expression for the commutator between two operators at the same point, $x = x'$. This is$$[\dot q(x,t),q(x,t)] = \delta(x-x').$$But you also need an expression for the commutator between two operators at different points, $|x-x'| = j\Delta x$. To get this you need to consider how $q(x,t)$ and $\dot q(x,t)$ vary over a small region around $x$, say from $x-\frac12\Delta x$ to $x+\frac12\Delta x$. If you expand $\dot q(x,t)$ and $q(x',t)$ into a Taylor series around $x$ up to second order, then you should find that$$[\dot q(x,t),q(x',t)] = \int_{-\frac12\Delta x}^{\frac12\Delta x} dy\delta(y+x-x') + O(\Delta x^2).$$The integral with the delta function is what you need, and the remainder is a higher order term which vanishes in the limit $\Delta x\to 0$.

1. What is the Canonical Commutation Relation (CCR)?

The Canonical Commutation Relation (CCR) is a fundamental concept in quantum mechanics and quantum field theory. It describes the relationship between two operators, such as position and momentum, and their corresponding observables, such as position and momentum measurements. The CCR is expressed as [A, B] = iℏ, where A and B are operators and ℏ is the reduced Planck's constant.

2. How is the CCR used in quantum mechanics?

The CCR is used to mathematically describe the uncertainty principle in quantum mechanics. It states that the product of the uncertainties of two complementary observables, such as position and momentum, cannot be smaller than the reduced Planck's constant. This means that the more precisely one observable is measured, the less precisely the other can be measured.

3. What is the significance of the CCR in quantum field theory?

The CCR is a key concept in quantum field theory, as it forms the basis for the quantization of fields. The field operators in quantum field theory satisfy the CCR, which allows for the creation and annihilation of particles. The CCR also plays a crucial role in the development of quantum field theories, such as the Standard Model, which describe the behavior of elementary particles.

4. What is the difference between the CCR and the Canonical Anticommutation Relation (CAR)?

While the CCR describes the commutation relation between operators, the CAR describes the anticommutation relation between operators. In quantum mechanics, fermions (particles with half-integer spin) follow the CAR, while bosons (particles with integer spin) follow the CCR. In quantum field theory, fermionic fields are quantized using the CAR, while bosonic fields use the CCR.

5. Are there any applications of the CCR outside of physics?

Yes, the CCR has applications in various fields, including signal processing, control theory, and quantum information theory. In signal processing, the CCR can be used to analyze the stability and performance of control systems. In quantum information theory, the CCR is used to study the uncertainty principle and its implications for quantum cryptography and quantum computing.

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