Understand of vacuum expectation

• wangyi
In summary, the vacuum expectation is the possibility amplitude of a photon turning into vacuum, but in the spontaneous symmetry breaking of gauge symmetry, it represents the strength of an electromagnetic field at vacuum. This can be understood by considering the perturbative part of quantizing small field excursions around the equilibrium value, which is taken to be zero in QED. In spontaneous symmetry breaking, the equilibrium value is not zero, so the vacuum expectation value includes both the equilibrium field value and the transition amplitude, which is 0. Therefore, the vacuum expectation value gives the equilibrium field value.

wangyi

Hello,
I have difficulty understanding the vacuum expectation:
consider <0|A_{mu}|0>, we can understand it as the
possibility ampitude of a photon turn into vacuum(although 0 in common),
but in the spontaneous of gauge symmetry, we should understand
<0|A_{mu}|0> as the strength of a electromagnetic field at vacuum.
How can I arrive at this explanation?

Calculate the ground state of a harmonic oscillator and yu will see it isn't zero.

Kruger said:
Calculate the ground state of a harmonic oscillator and yu will see it isn't zero.

But what I mean is not the zero-point energy, but the field strength itself,
for example, in electromagnetism the <state|A_{\mu}(x)|state>, does it mean the A value measured? or <state|E_{i}(x)|state>, does it mean the E value measured?

Generally speaking,those mathematical objects (called "transition probability amplitudes") are complex numbers,so they can't be measured in any way...Maybe their square modulus...

Daniel.

wangyi said:
Hello,
I have difficulty understanding the vacuum expectation:
consider <0|A_{mu}|0>, we can understand it as the
possibility ampitude of a photon turn into vacuum(although 0 in common),

That's the perturbative part: the quantisation of the small field excursions around their equilibrium value (which in QED, is taken to be the zero field).

but in the spontaneous of gauge symmetry, we should understand
<0|A_{mu}|0> as the strength of a electromagnetic field at vacuum.
How can I arrive at this explanation?

Well, in spontaneous symmetry breaking, the equilibrium value of the field is not zero. So the quantum particles (perturbatively) are the excursions around THAT equilibrium value ; the vacuum expectation value will then just give you both contributions: the equilibrium field value + plus the 1-quantum-to-zero-quantum transition amplitude. But as that last one is 0, you just get the equilibrium field value.

cheers,
Patrick.

What is vacuum expectation?

Vacuum expectation is a concept in quantum field theory that describes the average value of an operator in the vacuum state. It represents the expected value of a measurement taken on the vacuum state, which is the lowest energy state of a quantum system.

What is the importance of vacuum expectation in science?

Vacuum expectation is important in science because it helps us understand the behavior of quantum systems and their interactions. It also plays a crucial role in predicting and interpreting experimental results, particularly in high energy physics and quantum mechanics.

How is vacuum expectation calculated?

Vacuum expectation is calculated using mathematical formulas and equations, such as the Heisenberg uncertainty principle and the Schrödinger equation. These equations take into account the properties of the quantum system, such as its energy and momentum, to determine the expected value of a measurement.

What does vacuum expectation tell us about the vacuum state?

Vacuum expectation tells us that even though the vacuum state is the lowest energy state of a system, it still contains fluctuations and virtual particles that can arise and disappear spontaneously. These fluctuations contribute to the vacuum energy and can have measurable effects on the behavior of a system.

Can vacuum expectation be observed or measured?

No, vacuum expectation cannot be directly observed or measured. It is a theoretical concept that helps us understand the behavior of quantum systems. However, its effects can be indirectly observed through experiments and measurements of other physical properties, such as particle interactions and energy levels.