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LostConjugate
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In QFT after the Hamiltonian is normal ordered I guess when acting on the lowest energy state it annihilates it. So doesn't that mean the energy is zero now instead of 1/2?
G01 said:Try working out the zero point energy for the scalar field. You'll get:
[tex]<0|H|0>=V\int \frac{d^4k}{(2\pi)^4} \frac{1}{2}\hbar\omega(k)[/tex]
Even after you divide by the volume V, the energy of the vacuum still diverges because you are integrating [itex]\omega(k)=\sqrt{\vec{k}^2+m^2}[/itex] over an infinite number of modes.
So you may be able to cancel out the infinity arising from the spatial integral, but not the one arising from the k-space integral.
LostConjugate said:Ok so making the ground state 0 solves the problem. But what about the second state?
G01 said:You mean the 1 particle state?
The vacuum term is the same in the states with non zero particle number. Once you subtract it off the Hamiltonian once, you will get finite results for your non-zero particle states. For instance work out: [itex]<q|H'|q>=<q|H - H_o|q>[/itex] where [itex]H_o[/itex] is the groundstate energy defined above.
You will see that the ground state contribution cancels exactly with a term from [itex]<q|H|q>[/itex].
LostConjugate said:Sorry for my poor math knowledge.
Is this correct?
[tex] H-H_o = \int \frac{d^4 k}{(2\pi)^4} k_p a_p^\dagger a_p [/tex]Which is still infinite? Is |q> the same as the scalar field integral evaluated at p=1?
The equation H|0> = 0 represents the state of the zero-point energy of a quantum mechanical system. It means that the energy of the system is at its lowest possible value, also known as the ground state.
No, the equation H|0> = 0 does not mean that the zero-point energy is equal to zero. It simply represents the state of the energy at its minimum value. The actual value of the zero-point energy depends on the specific system and cannot be zero for all systems.
Zero-point energy is a concept in quantum mechanics that describes the lowest possible energy state of a system. It arises from the Heisenberg uncertainty principle, which states that there is always a minimum amount of energy associated with any physical system, even at absolute zero temperature.
Zero-point energy has various effects on physical systems, such as contributing to the stability of atoms and molecules, influencing the properties of materials, and causing the Casimir effect. It also plays a role in quantum fluctuations and the uncertainty principle.
Zero-point energy is a real concept in quantum mechanics. It has been observed and measured in various physical systems, and its effects have been confirmed through experiments and calculations. However, it is a difficult concept to understand and visualize, as it is not directly observable in our macroscopic world.