Understanding Weinberg's QFT Proof of Annihilation/Creation Operator Sum

In summary, Scott, Weinberg's statement is that any operator can be expressed as a sum of products of annihilation and creation operators.
  • #1
scottbekerham
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Hi . In weinberg's QFT book 1st volume he states that any operator can be expressed as a sum of products of annihilation and creation operators but I can't understand the proof . can simeone simplify this please? ( page 175)
 
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  • #2
hi scottbekerham! :smile:
scottbekerham said:
Hi . In weinberg's QFT book 1st volume he states that any operator can be expressed as a sum of products of annihilation and creation operators but I can't understand the proof . can simeone simplify this please? ( page 175)

he defines an operator O as something with a given value for each (ΦM,OΦN),

and then shows by induction how to get the required coefficients CNM

the inductive equation is the one ending "+ terms involving CNM with N < L etc" …

what do you not follow about that?
 
  • #3
Hi, Scott. Forget about Weinberg's proof for a while and picture this line of thought. Relativistic QFT comes nicely by generalizing single particle relativistic quantum mechanics which in turn generalizing the old quantum mechanics of Schroedinger, Dirac and Heisenberg. The latter is of course a mind-blowing extension of the classical mechanics of Hamilton. What can you say about an observable O in Hamilton mechanics ? It's defined on the phase space (here O becomes a function of p and q) by the property that, when evaluated on the surface of the solutions of Hamilton's equation, it's a mere numerical constant.

Going in reverse, p and q become operators in quantum mechanics, no longer variables of the phase space, so do the quantum mechanical observables become functions of the 'fundamental' operators p, q. But p and q in a single Hilbert space (actually in a RHS, but that's a finesse you won't need) can be linked to a and a^dagger. So the observables become functions of a and a^dagger. And now you go from a single HS to a Fock space and voila', you find Weinberg's statement.
 

FAQ: Understanding Weinberg's QFT Proof of Annihilation/Creation Operator Sum

What is Weinberg's QFT Proof of Annihilation/Creation Operator Sum?

Weinberg's QFT Proof of Annihilation/Creation Operator Sum is a mathematical proof in quantum field theory that shows how the annihilation and creation operators, which are used to describe the behavior of particles, can be summed to create a more elegant and simplified representation of quantum fields.

Why is Weinberg's QFT Proof important?

Weinberg's QFT Proof is important because it provides a rigorous mathematical foundation for understanding the behavior of particles in quantum field theory. It also allows for more efficient calculations and provides a deeper understanding of the underlying principles of quantum mechanics.

How does Weinberg's QFT Proof relate to the Standard Model of particle physics?

Weinberg's QFT Proof is a key component of the Standard Model of particle physics. The Standard Model is a theoretical framework that describes the fundamental particles and their interactions. Weinberg's proof helps to explain the behavior of these particles and how they interact with each other.

What are the implications of Weinberg's QFT Proof for future research?

Weinberg's QFT Proof has opened up new avenues for research in quantum field theory and particle physics. It has led to the development of new mathematical techniques and has provided a deeper understanding of the fundamental principles of quantum mechanics. This can potentially lead to new discoveries and advancements in our understanding of the universe.

Are there any criticisms of Weinberg's QFT Proof?

As with any scientific theory or proof, there have been some criticisms of Weinberg's QFT Proof. Some researchers have pointed out that the proof may not be applicable to all quantum field theories and that there may be some limitations to its use. However, the proof is widely accepted and has been extensively tested and validated through experiments and calculations.

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