# Why does the Phi-cubed theory have no ground state?

+ 1 like - 0 dislike
69 views

In the book of Sredinicki's, he claimed that the $\phi^3$ theory has no ground state, hence this is not a physical theory.
My question is that I can't see why this system has no ground state. And I don't understand either the explaination he gave. For example, what does "roll down the hill" really mean? What's the case for a harmonic oscillator pertuibed by a "q^3" term? Maybe it's better if someone can explain it using the quantum harmonic case.

+ 0 like - 0 dislike

Sredinicki means the exact $\phi^3$ theory. If you understand the exact solutions to the exact "harmonic oscillator stuffed with $x^3$ term", then you can get a gist of a field theory where the occupation numbers replace continuous $x$ of harminic oscillator problem.

If you do not understand the exact solutions to the exact "harmonic oscillator stuffed with $x^3$ term", then you may consider an infinite reflecting wall  $U(x)= 0,\; x<0,\; U(x)=+\infty,\; x\ge0$ as an example where the solutions are not localized. (There is a "ground state" with $E=0$, though.)

A "cubic oscillator" with big $g$ is similar to a slightly inclined wall (still reflecting), but with no "bottom" for negative-valued $x$, so there is no minimal $E_0$ for such a potential, no ground state, no localization.

answered Feb 24 by (112 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverflo$\varnothing$Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.