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i read in this paper (Spontaneous Symmetry Breaking as the Mechanism of Quantum Measurement by Michael Grady) that we are not allowed to consider the superposition of two vacuum states. i do not understand (page 4) why the superposition would violate a quantum property? what would be the problem of such a violation?

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  • $\begingroup$ To Dan Yang, thanks i like your answers. could you recall what is a local operator? is it on operator acting on functions with compact support? $\endgroup$ – Naima Mar 6 at 16:49
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    $\begingroup$ By "local operator", I mean an operator that has a nontrivial effect only within some bounded region of space. In particular, in quantum field theory, local observables (special local operators) associated with spacelike-separated regions must commute with each other. This is the microcausality or Einstein causality principle. $\endgroup$ – Chiral Anomaly Mar 14 at 2:18
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(In this answer, "different vacuum states" refers to different states that all have the the same minimum energy. This should not be confused with the "false vacuum" versus "true vacuum" language that is common in inflationary cosmology, for example.)

In principle, we can superpose two vacuum states, in the sense that quantum theory allows it. The catch is that we would never know that they were superposed — no experiment would ever be able to confirm it. That's because different vacuum states are more than just orthogonal to each other: they are prolifically orthogonal to each other, in the sense that no local operator can make them non-orthogonal to each other. In symbols, if $|a\rangle$ and $|b\rangle$ denote the two vacuum states, then we have $$ \langle a|X|b\rangle=0 \tag{0} $$ for all local operators $X$.

To see why this kind of "prolific orthogonality" implies that we can never experimentally confirm the existence of the superposition, work in the Heisenberg picture and suppose that we measure some sequence of local observables $A,B,C,...,Z$ associated with times $t_A,t_B,t_C,$ and so on. Let $A_1,A_2,...$ denote the projection operators associated with the possible outcomes of an $A$-measurement; let $B_1,B_2,...$ denote the projection operators associated with the possible outcomes of a $B$-measurement, and so on. According to Born's rule, if the system's configuration is described by the state-vector $|\psi\rangle$, then the "probability" (yeah, I know; it's just convenient slang) of obtaining the sequence of outcomes $A_i, B_j, C_k,...,Z_m$ is $$ p = \frac{\langle\hat\psi|\hat\psi\rangle}{\langle\psi|\psi\rangle} \tag{1} $$ with $$ |\hat\psi\rangle \equiv Z_m\cdots C_k B_j A_i|\psi\rangle. \tag{2} $$ The product $$ Z_m\cdots C_k B_j A_i \tag{3} $$ is a product of local operators associated with different (successive) times, and if we're using a quantum field theory that satisfies the usual causality principles, then this product is still a local operator — it is still a member of an algebra of operators associated with a bounded region of spacetime that encompasses the localization regions of all of the operators in the product [1]. Therefore, if we take $$ |\psi\rangle=|a\rangle+|b\rangle \tag{4} $$ (using un-normalized state-vectors so that we don't need to write explicit coefficients), then (1) becomes $$ p = \frac{\langle\hat a|\hat a\rangle}{\langle a|a\rangle} \frac{\langle a|a\rangle}{\langle\psi|\psi\rangle} + \frac{\langle\hat b|\hat b\rangle}{\langle b|b\rangle} \frac{\langle b|b\rangle}{\langle\psi|\psi\rangle}, \tag{5} $$ where each hat again denotes the result of applying the operator (3) to the unhatted version of the state-vector. Equation (5) says that no matter what sequence of measurements we consider, the "probabilities" of the possible outcome-sequences are always the same as if we had started with a mixed state given by a weighted sum of $|a\rangle\langle a|$ and $|b\rangle\langle b|$. In other words, we might as well have only $|a\rangle$ or only $|b\rangle$, even if we couldn't have predicted in advance which one we would have.

(Minor caveat: the preceding analysis assumed that the occurrence of the $B$-measurement is not contingent on the outcome of the preceding $A$-measurement, and likewise for the other measurements in the sequence. That assumption is not essential, but it simplifies the argument.)

By the way, such "prolifically orthogonal" states are sometimes said to belong to different superselection sectors. This is related to the fact that even a mixed state can always be represented as a single state-vector, by working in a direct-sum Hilbert space whose summands can't be mixed with each other by any observables. (This is the idea behind the GNS construction.)

The cited paper alludes to the cluster property, which is closely related to the idea that I just described. To see why they're related, use the abbreviation $$ \psi(X)\equiv\frac{\langle\psi|X|\psi\rangle}{\langle\psi|\psi\rangle}. \tag{6} $$ A state $|\psi\rangle$ is said to satisfy the cluster property if $$ \psi(XY)\approx\psi(X)\psi(Y) \tag{7} $$ whenever $X,Y$ are local operators separated by a sufficiently large spacelike distance (loosely speaking). If $|a\rangle$ and $|b\rangle$ both satisfy the cluster property, and if the condition (0) holds for all local observbles $X$, then $|\psi\rangle \equiv |a\rangle+|b\rangle$ is not expected to satisfy the cluster property (that's the connection between the two ideas). This is because any local operator $X$ satisfies $$ \psi(X)= a(X)\frac{\langle a|a\rangle}{\langle\psi|\psi\rangle} +b(X)\frac{\langle b|b\rangle}{\langle\psi|\psi\rangle} \tag{8} $$ (and likewise for $Y$ and for $XY$). Therefore, the product $\psi(X)\psi(Y)$ involves cross-terms with both $a$ and $b$, but $\psi(XY)$ does not have any such cross-terms, suggesting that $\psi$ can't satisfy the condition (7). This is analyzed more carefully in [2] and [3].

The impossibility of experimentally confirming a superposition of two vacuum states can be regarded as an extreme special case of the practical impossibility of confirming a superposition of outcomes after a high-quality measurement. This perspective and its relationship to spontaneous symmetry breaking is addressed in another post:

What does spontaneous symmetry breaking have to do with decoherence?

The relationship is also addressed in the paper cited by the OP. I haven't studied that paper carefully, but it seems at least roughly consistent with the perspective I used here: SSB is one example of a (naturally-occurring) measurement, and idealized SSB (infinite volume, infinite settling time, etc) is a mathematically-convenient limiting case that only occurs on paper, not in a laboratory. The statement that different vacuum states belong to different strict superselection sectors is a statement about that idealized limiting case.


[1] Haag (1996), Local Quantum Physics

[2] In the context of spin systems (like the Ising model): Section 23.3, "Order Parameter and Cluster Properties", of Zinn-Justin's book Quantum Field Theory and Critical Phenomena.

[3] In the context of relativistic QFT: Section 19.1 in Weinberg, The Quantum Theory of Fields, Volume II.

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