# Is non-relativistic quantum field theory equivalent with quantum mechanics?

+ 4 like - 0 dislike
100 views

Some books of many-body physics, e.g. A.L.Fetter and J.D.Walecka in Quantum theory of many-particle systems, claimed that at non-relativistic level, quantum mechanics (QM) and quantum field theory (QFT) are equivalent. They proved the second quantized operators $$T= \sum_{rs} \langle r | T | s \rangle a_r^{\dagger} a_s$$ $$V= \frac{1}{2} \sum_{rstu} \langle r s | T | t u \rangle a_r^{\dagger} a_s^{\dagger} a_u a_t$$

could obtain the same matrix elements as the "first-quantized" ones. Here $T$ and $V$ stand for kinetic and interaction operators, respectively.

However, some book, H. Umezawa et al Thermo field dynamics and condensed states in chapter 2, claimed that even at non-relativistic level, QFT is not equivalent with QM. They used a series of derivations (too long to present here, I may add a few steps if necessary), showed that Bogoliubov transformation with infinity space volume yields unitary inequivalent representations. In QM, all representations are unitary equivalent. Therefore, QM and non-relativistic QFT are not equivalent. However, as they said in p32

This might suggests that, in reality, the unitary inequivalence mentioned above may not happen because every system has a finite size. However, this point of view seems to be too optimistic. To consider a stationary system of finite size, we should seriously consider the effects of the boundary. As will be shown in later chapters, this boundary is maintained by some collective modes in the system and behaves as a macroscopic object with a surface singularity, which itself has an infinite number of degrees of freedom.

Nevertheless, Surface is an idealized concept. In reality, the boundary between two phases is a microscopic gradually changing of distribution of nuclei and electrons. My question is about, is the argument of surface singularity from Umezawa et al a pure academic issue? The academic issue here means if I have sufficient computational power, I compute all electrons and nuclei by quantum mechanics, I could very well reproduce the experimental results up to relativistic corrections.

P.S. The terminology "second-quantization" may not be appropriate, since we quantize the system only once. Nevertheless I could live with it.

This post imported from StackExchange Physics at 2014-08-11 14:53 (UCT), posted by SE-user user26143
 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$\hbar$ys$\varnothing$csOverflowThen 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.