# What causes the random decay of atoms in different interpretations?

What causes the "random" decay of atoms in different interpretations?

I know that atleast dBB and MWI needs to have a deterministic answer to this, so what is it?

Demystifier
Gold Member

Interesting question, but let me first slightly reformulate it into a more precise form:

What determines time at which an unstable atom will decay?

Standard QM says that nothing determines it, i.e. that only probability for a decay at given time is determined. By contrast, Bohmian QM should give a deterministic answer. But what that deterministic Bohmian answer is?

Admittedly, Bohmian QM cannot easily answer that question. But is it a problem of Bohmian QM? My answer is - no. More precisely, my claim is that it is not a problem of Bohmian QM per se, but actually a problem of standard QM that reflects in Bohmian QM as well.

Let me explain. In Bohmian QM there is a general theorem saying that Bohmian QM makes the same measurable statistical predictions as standard QM whenever the predictions of standard QM are unambiguous. Indeed, in standard QM, the probability distribution is unambiguous for any observable (such as position, momentum, energy, spin, etc.) given by an operator in the Hilbert space. But the problem is that time in QM is not an operator. Or at least, the time operator in QM is not unambiguously defined. The issue of time in QM is a controversial subject even within standard QM. There are several proposals for the solution of it, but none of them is a "standard" one.

So, without an unambiguous probabilistic description of decay time in standard QM, one shouldn't expect an unambiguous answer in Bohmian QM. Instead, it seems reasonable to choose one specific (even if not generally accepted) approach in standard QM and then explore the Bohmian variant of it as well. This is what I will do in the next post ...

Demystifier
Gold Member

... So let me briefly explain one approach (in both standard and Bohmian QM) that seems most promissing to me. An interested reader can find more details in
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595-602]
http://xxx.lanl.gov/abs/1002.3226 [Int. J. Quantum Inf. 9 (2011) 367-377]

The essence of this approach is to treat time on an equal footing with space. Space is treated as in the standard approach, while time is treated just as one additional (fourth) dimension. This approach is relativistic in spirit, but a non-relativistic limit of it is also different from the usual treatment of time in QM. In fact, it can be thought of as a generalization of usual QM, in the sense that wave functions now live in an extended Hilbert space - a space of functions of both x and t.

With this approach, the usual rules of thumb on decay probabilities in standard QM now can be derived from first principles, just as for any other quantum observable. But how the Bohmian deterministic approach determines time at which the decay will happen? As usual in Bohmian QM, it all depends on the initial particle positions. However, now the "initial position" has a somewhat different meaning. While in the usual Bohmian QM the initial position is the set
x(t=0), y(t=0), z(t=0),
which clearly does not treat time on an equal footing with space, in the approach I am talking about the initial condition is the set
x(s=0), y(s=0), z(s=0), t(s=0)
where s is a relativistic-scalar parameter that parameterizes the trajectory. This parameter can be thought of as a generalized proper time. In this approach time t is also one of the hidden variables that may not be known in advance, which in essence is why the time of an event (such as a decay) looks random. As discussed in more detail in the papers above, the final statistical predictions coincide with those of standard QM.

Demystifier